On the bad reduction of certain U(2, 1) Shimura varieties

TL;DR

This paper investigates the structure, singularities, and stratifications of three related Picard modular surfaces in characteristic p, focusing on their components and morphisms, especially around superspecial points.

Contribution

It provides a detailed analysis of the components, singularities, and stratifications of Picard modular surfaces and introduces a foliation on the blow-up at superspecial points.

Findings

Classification of components and singularities of the surfaces.

Description of stratifications and their behavior under morphisms.

Introduction of a foliation on the blow-up at superspecial points.

Abstract

Let $E$ be a quadratic imaginary field and let $p$ be a prime which is inert in $E.$ We study three types of Picard modular surfaces in positive characteristic $p$ and the morphisms between them. The first Picard surface, denoted $S$, parametrizes triples $(A,\phi,\iota)$ comprised of an abelian threefold $A$ with an action $\iota$ of the ring of integers $\mathcal{O}_{E}$, and a principal polarization $\phi$. The second surface, $S_{0}(p)$, parametrizes, in addition, a suitably restricted choice of a subgroup $H\subset A[p]$ of rank $p^{2}$. The third Picard surface, $\widetilde{S}$, parametrizes triples $(A,\psi,\iota)$ similar to those parametrized by $S$, but where $\psi$ is a polarization of degree $p^{2}$. We study the components, singularities and naturally defined stratifications of these surfaces, and their behaviour under the morphisms. A particular role is played by a foliation we define on the blow-up of $S$ at its superspecial points.

Equations

$$S_0(p)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathcal{T}\cong S_0(p){\times }_{\tilde{S}}{S}_0(p).$$

$$k[[x]]\hookrightarrow k[[x,y]]/(xy),$$

$$k[[x]]\hookrightarrow k[[x,y]]/((x^p-y)(x-y^p)),$$

$$X^{(p)}=S{\times }_{{\Phi }_S,S}X$$

$$({\Phi }_S\times 1)\circ F{r}_{X/S}={\Phi }_X.$$

$$(u,v)=\,^{t}\bar{u}\left(\begin{array}[]{ccc}&&1\\ &1\\ 1\end{array}\right)v,$$

$$\mathcal{D}=\left\{(z_0:z_1:z_2)|{\overline{z}}_0{z}_2+{\overline{z}}_1{z}_1+{\overline{z}}_2{z}_0<0\right\},$$

$$S_K(\mathbb{C})\simeq \mathbf{G}(\mathbb{Q})\setminus \mathbf{G}(\mathbb{A})/K\simeq \mathbf{G}(\mathbb{Q})\setminus [\mathcal{D}\times \mathbf{G}({\mathbb{A}}_f)/K_f].$$

$${\Lambda }_0=\Lambda \otimes {\mathbb{Z}}_p={⟨e_0,e_1,e_2⟩}_{{\mathcal{O}}_p}.$$

$${\Lambda }_1={⟨p{e}_0,e_1,e_2⟩}_{{\mathcal{O}}_p}.$$

$${\Lambda }_2={⟨p{e}_0,p{e}_1,e_2⟩}_{{\mathcal{O}}_p}.$$

$$K_p^1=K_p^0\cap {\tilde{K}}_p^0.$$

$$\pi :S_0(p)\to S,\tilde{\pi }:S_0(p)\to \tilde{S}$$

$$I^{(i)}=\{f\in I|\forall a\in {\kappa }^{\times },\iota (a{)}^{\ast }(f)={\omega }^i(a)f\}.$$

$$F:M^{(p)}\to M,V:M\to M^{(p)},$$

$$0\to H_1\to H\stackrel{\mathrm{Fr}}{\to }{H}_1^{(p)}\to 0.$$

$$\mathrm{Lie}(H)=M[V]=M$$

$${\iota }^D(a)=\iota (a{)}^t,$$

$$⟨x,{\iota }^D(a)y⟩=⟨\iota (a)x,y⟩\in (R^{\prime }{)}^{\times }.$$

$$⟨,⟩:\Lambda /N\Lambda \times \Lambda /N\Lambda \to \mathbb{Z}/N\mathbb{Z}$$

$$\pi :{\mathcal{S}}_0(p)\to \mathcal{S},\tilde{\pi }:{\mathcal{S}}_0(p)\to \tilde{\mathcal{S}}$$

$$0\to H\to A\to A^{\prime }\to 0$$

$$\begin{array}[]{ccccccccc}0&\to&M(A)&\to&M(A^{\prime})&\to&M(H)&\to&0\\ &&\downarrow V&&\downarrow V&&\downarrow V\\ 0&\to&M(A)^{(p)}&\to&M(A^{\prime})^{(p)}&\to&M(H)^{(p)}&\to&0\end{array}$$

$$\begin{array}[]{cccccc}0&\to&M(H)[V]&\to&M(A)^{(p)}/VM(A)&\to\\ &&\|&&\|\\ 0&\to&{\rm Lie}(H)&\to&{\rm Lie}(A)&\to\end{array}$$

$$\begin{array}[]{cccccc}\to&M(A^{\prime})^{(p)}/VM(A^{\prime})&\to&M(H)^{(p)}/VM(H)&\to&0\\ &\|&&\|\\ \to&{\rm Lie}(A^{\prime})&\to&M(H)^{(p)}/VM(H)&\to&0\end{array}$$

$$0\to M(H)[V]\to M(H)\stackrel{V}{\to }M(H{)}^{(p)}\to M(H{)}^{(p)}/VM(H)\to 0$$

$$p\varphi =f^t\circ \psi \circ f.$$

$${\underline{A}}^{\prime }=\underline{A}/H,J=A[p]/H,$$

$$⟨a⟩(A,\varphi ,\iota ,\eta )=(A,\varphi ,\iota ,\eta \circ a)=(A,\varphi ,\iota ,\iota (a)\circ \eta ).$$

Full text

Dedicated to V. Kumar Murty on the occasion of his 60th birthday

1. 1 Three integral models with Parahoric level structure

2. 1.1 Shimura varieties

3. 1.2 The moduli problems

4. 1.2.1 The Raynaud condition

5. 1.2.2 The moduli problem $(S)$

6. 1.2.3 The moduli problem $(\widetilde{S})$

7. 1.2.4 The moduli problem $(S_{0}(p))$

8. 1.2.5 The maps between the integral models

9. 1.2.6 The moduli problem $(\widetilde{S}_{0}(p))$

10. 1.2.7 Diamond operators

11. 1.3 Translation into the language of Rapoport and Zink

12. 1.4 Modular curves on the Picard modular surface

13. 1.4.1 Embedding the modular curve

14. 1.4.2 Endomorphism rings of $\mathbb{\overline{F}}_{p}$ points of $\mathscr{S}$

15. 2 The structure of the special fiber of $\mathscr{S}$

16. 2.1 Stratification

17. 2.2 The tangent bundle of $S$

18. 2.2.1 The special line sub-bundle $TS^{+}$

19. 2.2.2 A characterization in terms of generalized Serre-Tate coordinates

20. 2.3 The blow up of $S$ at the superspecial points

21. 3 Local structure of the three integral models

22. 3.1 Raynaud’s classification

23. 3.2 The completed local rings

24. 3.2.1 Generalities on local models

25. 3.2.2 The standard model

26. 3.2.3 The Hodge filtration

27. 3.2.4 Deformations

28. 3.3 Computations

29. 3.3.1 Local model diagrams

30. 3.3.2 First example

31. 3.3.3 Second example

32. 4 The global structure of $S_{0}(p)$

33. 4.1 The $\mu$-ordinary strata

34. 4.1.1 Lots of Frobenii

35. 4.1.2 The $\mu$-ordinary strata

36. 4.1.3 A lemma on ramification

37. 4.1.4 The unramified direction of $\overline{\pi}_{et}$

38. 4.2 The gss strata

39. 4.2.1 The $\mathbb{P}^{1}$-bundles

40. 4.2.2 Proof of Theorem 4.5

41. 4.2.3 A closer look at Example 3.3.2

42. 4.3 The ssp strata

43. 4.3.1 The superspecial combs

44. 4.3.2 The maps to $S^{\#}$

45. 4.3.3 How embedded modular curves meet $F_{x}$

46. 5 The structure of $\widetilde{S}$

47. 5.1 The global structure of $\widetilde{S}$

48. 5.1.1 Flatness of $\widetilde{\pi}$

49. 5.1.2 The fibers of $\widetilde{\pi}$

50. 5.1.3 The stratification of $\widetilde{S}$

51. 5.2 Analysis of $\widetilde{\pi}$

52. 5.2.1 Analysis of $\widetilde{\pi}$ along the $\mu$-ordinary strata

53. 5.2.2 Analysis of $\widetilde{\pi}$ along the curves $Z_{et}$ and $Z_{m}$

54. 5.2.3 Analysis of $\widetilde{\pi}$ along the gss screens $Y_{gss}$

55. 6 Appendix

56. 6.1 The classification of the gss Dieudonné modules

57. 6.2 The quotient $A/H$ for gss $A$

Introduction

Let $E$ be a quadratic imaginary field and let $p$ be a prime which is inert in $E.$ This paper is concerned with the detailed study of three types of Picard modular surfaces in positive characteristic $p$ and the morphisms between them. Deferring precise definitions to the body of the paper, the first Picard surface, denoted $S$, parametrizes triples $(A,\phi,\iota)$ comprised of a certain abelian threefold $A$ with an action $\iota$ of the ring of integers $\mathcal{O}_{E}$, and a principal polarization $\phi$. Unlike the other two, $S$ is smooth. The second surface, $S_{0}(p)$, parametrizes, in addition, a suitably restricted choice of a subgroup $H\subset A[p]$ of rank $p^{2}$. The third Picard surface, $\widetilde{S}$, parametrizes triples $(A,\psi,\iota)$ similar to those parametrized by $S$, but where $\psi$ is a polarization of degree $p^{2}$. There are natural morphisms providing us with a diagram

[TABLE]

From another perspective, there are three Shimura varieties associated with the unitary group of $E$ of signature (2,1), having parahoric level structure at $p.$ The above mentioned moduli spaces are the special fibers at $p$ of the integral models of these Shimura varieties, studied by Rapoport and Zink in [Ra-Zi].

Before describing the main results of this article, we provide some background, context and motivation. Picard modular surfaces appear in many places in the literature; the book by Langlands and Ramakrishnan [La-Ra] provides a strong motivation for their study as a test case for the Langlands conjectures on modularity of $L$-functions, as well as a guide to the literature at the time. The local structure at $p$ of $S_{0}(p)$ and related moduli spaces was studied in Bellaïche’s thesis [Bel], and later in the work of Bültel-Wedhorn [Bu-We] and Koskivirta [Kos], where the authors applied it to lifting problems of Picard modular forms, Galois representations, and congruence relations for Hecke operators. However, the global structure of $S_{0}(p)$ and of the map $S_{0}(p)\to S$ remained opaque. Thus, one of our original motivations was to make this global structure precise.

Unlike $S_{0}(p),$ there is little information in the literature on $\widetilde{S}$, or in general on moduli spaces of abelian varieties with non-separable polarizations. The main examples we are aware of are [Cri, dJ1, Nor, N-O], and they tend to exhibit rather pathological phenomena. It is desirable to have additional examples available, and indeed $\widetilde{S}$, in contrast to loc. cit., has proven to be extremely well-behaved.

Our main reason for studying the three Picard modular surfaces, was however different. Motivated by questions on the canonical subgroup, or by the search for a geometric proof of the congruence relation (as in [Bu-We, Kos]), it is desirable to have a surface parametrizing tuples $(A,\phi,\iota,H)$, where $H$ is a finite flat subgroup scheme which may reduce mod $p$ to the kernel of Frobenius. As this kernel has rank $p^{3}$ and in characteristic [math] the rank of a $p$-primary $\mathcal{O}_{E}$-subgroup must be an even power of $p,$ such a surface does not exist. To remedy the situation, one is forced to consider a moduli space as above, but where $H$ is now of rank $p^{6}$. In the context of modular curves this is akin to passing from $X_{0}(p)$ to $X_{0}(p^{2})$; a process which is, of course, unnecessary for modular curves, but would be required for many Shimura varieties.

It turns out that it is beneficial to modify the moduli problem somewhat and following [dJ2] to consider a filtration of $H$ as part of the datum. That is, (roughly) the following data: $(A,\phi,\iota,H_{0}\subseteq H)$, where $(A,\phi,\iota,H_{0})$ is an object parametrized by $S_{0}(p)$ and $H$ is a suitable rank $p^{6}$ finite flat subgroup scheme. We call this moduli problem $\mathcal{T}$, and one of our initial observations is that

[TABLE]

In characteristic $0,$ this surface is finite flat of degree $(p+1)(p^{3}+1)$ over $S,$ and represents the Hecke operator $T(p).$ This, therefore, motivated both the introduction of $\widetilde{S}$ and the study of the morphism $\widetilde{\pi}$. The study of the moduli space $\mathcal{T}$ will be carried out in a subsequent paper. Nonetheless, the foundations are laid down here.

While studying the three moduli spaces $S,S_{0}(p)$ and $\widetilde{S}$, we discovered a new interesting phenomenon. The generic stratum of $S$ in characteristic $p$ parametrizes $\mu$-ordinary abelian threefolds. Although their $p$-divisible groups are all isomorphic, studying their cotangent spaces we were able to distinguish in the tangent space of $S$ a certain “foliation”, amounting in this very simple example to a line sub-bundle closed under the operation of raising to power $p$ (see §2.2). The link between the cotangent space of the universal abelian variety and that of $S$ is supplied by the Kodaira-Spencer map. This foliation extends to the general supersingular locus of $S,$ but fails to extend, in a way made precise, to the superspecial points there. Moreover, we found two other ways to characterize it: the first, as the foliation of “unramified directions” (in the sense of [Ru-Sh]) for a map $\overline{\pi}:S_{0}(p)^{(p)}\to S$ derived from the map $\pi$ (Theorem 4.4). The second, in terms of Moonen’s generalized Serre-Tate coordinates [Mo] (Proposition 2.4). Shimura curves embedded in $S,$ as well as the supersingular curves in $S,$ are integral curves of this foliation (Theorem 2.3). Does it have any other global integral curves? We expect this new phenomenon to generalize to other Shimura varieties of PEL type whose generic stratum is $\mu$-ordinary but not ordinary; cf. our forthcoming paper [dS-G3] where such a foliation is studied for unitary Shimura varieties of arbitrary signatures.

A summary of the results

We now describe briefly the content of this paper. Chapter 1 reviews the three Shimura varieties and their integral models. We explain the precise relation between the moduli problem with parahoric level structure as in [Ra-Zi] and the Raynaud condition appearing in [Bel]. The last section reviews the embeddings of modular curves and Shimura curves in the Picard modular surface.

Chapter 2 deals with the Picard modular surface $S$, where the level at $p$ is a hyperspecial maximal compact. The mod $p$ fiber is smooth, and its stratification was studied by Vollaard in [Vo]. It consists of three strata. The dense open stratum $S_{\mu}$ parametrizes $\mu$-ordinary abelian threefolds. Its complement $S_{ss}$ parametrizes supersingular ones, and consists (at least when the tame level $N$ is large, depending on $p$) of Fermat curves of degree $p+1,$ intersecting transversally at their $\mathbb{F}_{p^{2}}$-rational points. These intersection points support superspecial abelian threefolds (isomorphic, not only isogenous, to a product of supersingular elliptic curves), and constitute the third stratum $S_{ssp}.$ The non-singular locus of the curve $S_{ss}$ supports supersingular, but not superspecial, abelian threefolds, and is denoted $S_{gss}.$ This is the intermediate stratum. The number of its irreducible components was determined in [dS-G1] using intersection theory on $S$ and a secondary Hasse invariant constructed there. It turns out to be related to the second Chern number of $S,$ and via a result of Holzapfel, expressible as an $L$-value. Our contribution to the study of $S$ in the present paper is: (a) We introduce the foliation $TS^{+}$ in the tangent bundle of $S$, outside $S_{ssp}$, and prove the results to which we alluded above; (b) We introduce the blow-up $S^{\#}$ of $S$ at $S_{ssp}$ and give it a modular interpretation. It has the advantage that the irreducible components of $S_{ss}$ become, after blowing up, disjoint non-singular Fermat curves (even when $N$ is small), i.e. all their intersections, including self-intersections, are resolved. The exceptional divisor at every blown-up point $x$ is a projective line $E_{x}$ defined over $\mathbb{F}_{p^{2}}.$ The components of $S_{ss}$ intersect $E_{x}$ at points $\zeta$ satisfying $\zeta^{p+1}=-1.$ Embedded Shimura curves, on the other hand, intersect $E_{x}$ at $\mathbb{F}_{p^{2}}$-rational points satisfying $\zeta^{p+1}\neq-1.$ The proofs of these results will have to wait until Theorem 4.11 and §4.3.3.

Chapter 3 is based on chapter III of Bellaïche’s thesis [Bel] and describes the local models for the completed local rings of the three Shimura varieties, at any point of the special fiber. We are nevertheless interested not only in the completed local rings per se, but in the maps between them. The theory of local models yields these maps only modulo $p$th powers of the maximal ideal. This is evident already in the case of the germ of the map $X_{0}(p)\to X$ between two modular curves, with and without $\Gamma_{0}(p)$-level structure, at a supersingular point. In this “baby case” the map between the local models is

[TABLE]

which is not even flat. The correct map, however, is known ever since Kronecker to be

[TABLE]

which is finite flat of degree $p+1$. Similar but more serious problems arise when we study the maps between the completed local rings of our three Picard surfaces. Luckily, a general theorem of Rudakov and Shafarevich [Ru-Sh] on the local structure of inseparable maps between smooth surfaces, allows us to give a partial answer to our question. In essence, it allows us to determine the maps between the completed local rings of the analytic branches through any given point. Once again, results of this type have to await the study of $S_{0}(p)$ and $\widetilde{S}$ in subsequent chapters, where we relate them also to the foliation $TS^{+}$ mentioned above.

Chapter $4$ is the longest, and deals with the Picard surface $S_{0}(p)$ of Iwahori level structure, and the map $\pi$ from $S_{0}(p)$ to $S.$ We caution that $\pi$ is neither finite nor flat. The special fiber of $S_{0}(p)$ consists of vertical and horizontal components intersecting transversally. There are two horizontal components, multiplicative and étale. The multiplicative component maps under $\pi$ isomorphically onto $S^{\#}$. The map from the étale component is purely inseparable of degree $p^{3}$ and factors through Frobenius. The factored map $\overline{\pi}_{et}$ is inseparable of degree $p,$ and we show that its “field of unramified directions” is just the foliation $TS^{+}$, which was defined before intrinsically on $S.$ The vertical components of $\pi$ are $\mathbb{P}^{1}$-bundles over Fermat curves, which we call the “supersingular screens”. Above each superspecial point $x\in S_{ssp}$ lies in $S_{0}(p)$ a “comb”, whose base $F_{x}$ is a $\mathbb{P}^{1}$ along which the two horizontal sheets of $S_{0}(p)$ meet, and whose “teeth” $G_{x}[\zeta]$ belong to the supersingular screens. For a more precise description we refer to Theorems 4.1, 4.5 and 4.11 and their corollaries.

Chapter 5 deals with $\widetilde{S}$ and the map $\widetilde{\pi}$. Unlike $\pi,$ this map is finite flat of degree $p+1.$ Here again there are horizontal and vertical components. This time $\widetilde{\pi}$ is an isomorphism on the étale component of $S_{0}(p)$ and purely inseparable of degree $p$ on the multiplicative component. The maps $\pi$ and $\widetilde{\pi}$ allow us to go back and forth between $S$ and $\widetilde{S}$ and produce maps that we are able to analyze easily in light of the modular interpretation. On the vertical components of $S_{0}(p)$ (the supersingular screens) the map $\widetilde{\pi}$ is pretty intricate. We collect some results on it in the last section of Chapter 5, but leave some other questions unanswered.

The appendix contains some ugly but unavoidable computations with Dieudonné modules, that would have interrupted the presentation, had they been left where needed.

Deformation theory of $p$-divisible groups clearly is a central tool in this work. Unfortunately, there are at least three traditional approaches to it: Grothendieck’s theory of crystals, contravariant Dieudonné theory, and covariant Dieudonné-Cartier theory (not counting displays, $p$-typical curves etc.). We made every effort to remain faithful to the language and notation used by the various references cited by us. This resulted, however, in a mixture of the three approaches. A very useful guide, and a dictionary between the various languages, can be found in the appendix to [C-C-O].

Notation

• •

If $A$ is an abelian scheme over a base $S,$ $A^{t}$ denotes its dual abelian scheme.

• •

If $H$ is a finite flat group scheme over a base $S,$ $H^{D}$ denotes its Cartier dual.

• •

If $S$ is a scheme over $\mathbb{F}_{p}$ we denote by $\Phi_{S}:S\to S$ the absolute Frobenius morphism of $S.$ If $X\to S$ is any scheme, we denote by $X^{(p)/S},$ or simply by $X^{(p)},$ if no confusion may arise, the fiber product

[TABLE]

and by $Fr_{X/S}:X\to X^{(p)}$ the unique morphism over $S$ such that

[TABLE]

• •

If $A$ is an abelian scheme over $S$ then ${\rm Fr}=Fr_{A/S}:A\to A^{(p)}$ is an isogeny (the Frobenius of $A$). The Verschiebung ${\rm Ver}:A^{(p)}\to A$ is the isogeny dual to the Frobenius of $A^{t}$.

• •

If $\lambda:A\to A^{t}$ is a polarization of an abelian scheme $A$ and $K=\ker\lambda$, we denote by $e_{\lambda}:K\times K\to\mathbb{G}_{m}$ the Mumford pairing on $K$. If $\lambda=n\phi$ where $\phi$ is a principal polarization, then $e_{\lambda}$ is Weil’s $e_{n}$-pairing associated with $\phi.$

• •

$E$ is a quadratic imaginary field, $\mathcal{O}_{E}$ its ring of integers, $p$ a prime that remains inert in $E,$ $\kappa=\mathcal{O}_{E}/p\mathcal{O}_{E}$ and $\mathcal{O}_{p}$ is the completion of $\mathcal{O}_{E}$ at $p.$ We write $\sigma$ for the non-trivial automorphism of $E,$ extended to $\mathcal{O}_{p}.$

• •

If $R$ is an $\mathcal{O}_{p}$-algebra we denote by $\Sigma$ the given homomorphism $\mathcal{O}_{p}\to R$ and $\overline{\Sigma}=\Sigma\circ\sigma.$

• •

If $G$ is a commutative group scheme over a base $S$ we denote by $\mathcal{O}_{E}\otimes G$ the $S$-group scheme representing the functor $S^{\prime}\mapsto\mathcal{O}_{E}\otimes_{\mathbb{Z}}G(S^{\prime})$. It has an obvious $\mathcal{O}_{E}$ action.

• •

If $X$ is a non-singular algebraic variety over a field $k$ we denote its tangent bundle by $TX$. The fiber of $TX$ at $x\in X(k)$ (the tangent space at $x$) will be denoted by $T_{x}X=TX|_{x}.$

• •

If $X$ is any scheme we denote by $X^{red}$ the same underlying space, equipped with the reduced induced subscheme structure.

Acknowledgments: We thank Ben Moonen and George Pappas for helpful discussions. We are grateful to the research institutes IHES, Bures-sur-Yvette and MFO, Oberwolfach, where part of the research on this paper has been done, for their hospitality.

1. Three integral models with Parahoric level structure

1.1. Shimura varieties

Let $E$ be a quadratic imaginary field. Let $\Lambda=\mathcal{O}_{E}^{3}$, equipped with the hermitian form

[TABLE]

which is of signature $(2,1)$ over $\mathbb{R}$. We denote by $e_{0},e_{1},e_{2}$ the three standard basis vectors. Let $\boldsymbol{G}$ be the group of unitary similtudes $GU(\Lambda,(,))$, regarded as a linear algebraic group over $\mathcal{\mathbb{Z}}$. The Shimura varieties in the title will be associated with $\boldsymbol{G}$. More precisely, $G_{\infty}=\boldsymbol{G}(\mathbb{R})$ acts by projective linear transformations on $\mathbb{P}^{2}(\mathbb{C})$. The bounded symmetric domain

[TABLE]

biholomorphic to the unit ball in $\mathbb{C}^{2}$, is preserved by $G_{\infty}$, which acts on it transitively. Denote by $K_{\infty}$ the stabilizer of the “center” $(-1:0:1).$ For any compact open subgroup $K_{f}\subset\boldsymbol{G}(\mathbb{A}_{f})$ we put $K=K_{\infty}K_{f}\subset\boldsymbol{G}(\mathbb{A})$.

The Shimura variety $S_{K}$ is a quasi-projective variety over $E$ whose complex points are identified, as a complex manifold, with

[TABLE]

Fix an odd prime $p$ which is inert in $E,$ and let $N\geq 3$ be an integer such that $p\nmid N$. Let $\kappa=\mathcal{O}_{E}/p\mathcal{O}_{E}$ and denote by $\mathcal{O}_{p}$ the ring of integers in the completion $E_{p}.$ Assume that $K_{f}=K_{p}K^{p}$ where $K^{p}\subset\boldsymbol{G}(\mathbb{A}_{f}^{p})$ is the principal level subgroup of level $N,$ and $K_{p}\subset G_{p}=\boldsymbol{G}(\mathbb{Q}_{p})$.

In this paper we are interested in three choices of $K_{p}.$ As $p$ is inert in $E,$ $G_{p}$ is non-split, and its semi-simple rank is 1. Its Bruhat-Tits building is a biregular tree of bi-degree $(p^{3}+1,p+1)$. The vertices of degree $p^{3}+1$ are stabilized by hyperspecial maximal compact subgroups of $G_{p}$, which are all conjugate to $K_{p}^{0}=\boldsymbol{G}(\mathbb{Z}_{p}).$ This subgroup is the stabilizer of the standard self-dual lattice

[TABLE]

The vertices of degree $p+1$ are stabilized by special, but not hyperspecial, maximal compact subgroups, which are all conjugate to the stabilizer $\widetilde{K}_{p}^{0}$ of the lattice

[TABLE]

Note that this is also the stabilizer of $p^{-1}\Lambda_{2}$, the dual lattice with respect to the hermitian pairing, where

[TABLE]

We call the vertices of degree $p^{3}+1$ vertices of type (hs) and the ones of degree $p+1$ of type (s). The vertices $v_{0}$ and $\widetilde{v}_{0}$ corresponding to $K_{p}^{0}$ and $\widetilde{K}_{p}^{0}$ are called the *standard *vertices of the respective types. The oriented edge $(v_{0},\widetilde{v}_{0})$ is then stabilized by the standard Iwahori subgroup

[TABLE]

We denote by $S$ (resp. $\,\widetilde{S}$, resp. $\,S_{0}(p)$) the Shimura variety over $E$ of level $K_{f}=K_{p}K^{p},$ where $K^{p}$ is as above (of full tame level $N$) and $K_{p}=K_{p}^{0}$ (resp. $\,\widetilde{K}_{p}^{0},$ resp. $\,K_{p}^{1}$). The following result is well-known.

Proposition 1.1**.**

The Shimura varieties S, $\widetilde{S}$ and $S_{0}(p)$ are non-singular quasi-projective surfaces over $E$ and the natural maps

[TABLE]

are finite étale of degrees $p^{3}+1$ and $p+1$ respectively.

We denote by $\mathscr{S}$ (resp. $\,\mathscr{\widetilde{S}},$ resp. $\,\mathscr{S}_{0}(p)$) the integral models of these varieties over $\mathcal{O}_{p}$ constructed in chapter 6 of [Ra-Zi]. They are of relative dimension $2,$ $\mathscr{S}$ is smooth over $\mathcal{O}_{p}$, but the other two are not. The relative surface $\mathscr{S}$ is the integral model of the Picard modular surface which has been studied in detail by Vollaard [Vo] §§4-6. See [dS-G1] for related results. The surface $\mathscr{S}_{0}(p)$ has been studied to some extent in Bellaïche’s thesis [Bel]. Previous to this paper, little was known about $\mathscr{\widetilde{S}}$, apart from the general facts that follow from [Ra-Zi]. We review these three integral models in the next section.

From a general theorem of Görtz [Gö], or from the computations of the local models cited in §3.2, it follows that all three integral models are flat over $\mathcal{O}_{p}$, and their special fibers are reduced. As we shall later show, they are also regular.

1.2. The moduli problems

1.2.1. The Raynaud condition

Let $R$ be a commutative $\mathcal{O}_{p}$-algebra and $H$ a finite flat group scheme over $R$ of rank $p^{2}.$ Assume that we are given a ring homomorphism $\iota:\mathcal{O}_{E}\rightarrow{\rm{\rm End}}_{R}(H)$, and that $H$ is killed by $p,$ or, equivalently, $\iota$ factors through the field $\kappa=\mathcal{O}_{E}/p\mathcal{O}_{E}.$ Locally on $Spec(R),$ $\mathcal{O}(H)=A$ is free of rank $p^{2}$; the zero section of $H$ is given by an $R$-homomorphism $\epsilon:A\rightarrow R$ whose kernel $I,$ the augmentation ideal, is free of rank $p^{2}-1.$ Letting $a\in\kappa^{\times}$ act on $A$ via $\iota(a)^{*},$ this becomes a group action, which preserves $I$. Let $\omega:\kappa^{\times}\to\mathcal{O}_{p}^{\times}\to R^{\times}$ be the Teichmüller character, and for $1\leq i\leq p^{2}-1$ let

[TABLE]

Thanks to the fact that $p^{2}-1$ is invertible in $R$, these are distinct $R$-submodules, and $I$ is their direct sum. Following [Bel, Ray], we call $H$ *Raynaud *if each $I^{(i)}$ is free of rank 1 over $R$. The following facts are easily checked.

• •

Let $R\to R^{\prime}$ be any base change. Then if $H$ is Raynaud, so is $H\times_{Spec(R)}Spec(R^{\prime})$.

• •

The converse holds if $Spec(R)$ is connected. In particular, it is enough to check then the Raynaud condition at one geometric point.

• •

The constant group scheme $\mathcal{O}_{E}\otimes\mathbb{Z}/p\mathbb{Z}$ and its dual $\mathcal{O}_{E}\otimes\mu_{p}$ are Raynaud.

It follows from the three properties that étale and multiplicative (dual to étale) group schemes are automatically Raynaud.

Assume now that $R=k$ is a perfect field containing $\kappa.$ Let $M=M(H)$ be the covariant Dieudonné module111We adhere to the conventions of [C-C-O], Appendix B.3. Our $M(H)$ is denoted there $M_{*}(H).$ $F$ and $V$ can be regarded also as $\sigma$ or $\sigma^{-1}$-linear maps on $M$. Recall that $V$ is induced by ${\rm Fr}:H\to H^{(p)}$ and $F$ is induced by ${\rm Ver}:H^{(p)}\to H.$ of $H.$ Since $H$ is killed by $p,$ $M$ is a $2$-dimensional vector space over $k,$ equipped with linear maps

[TABLE]

where $M^{(p)}=k\otimes_{\sigma,k}M$ and $\sigma(x)=x^{p}$ is the Frobenius on $k.$ The action of $\kappa$ on $H$ induces an action of $\kappa$ on $M$; we let $M(\Sigma)$ be the subspace on which $\kappa$ acts through the natural embedding $\Sigma:\kappa\hookrightarrow k$, and $M(\overline{\Sigma})$ the subspace on which it acts via $\overline{\Sigma}=\sigma\circ\Sigma.$ Then $M=M(\Sigma)\oplus M(\overline{\Sigma}).$ Note that $(M^{(p)})(\Sigma)=(M(\overline{\Sigma}))^{(p)}$ and vice versa. We call $M$ balanced if both $M(\Sigma)$ and $M(\overline{\Sigma})$ are $1$-dimensional.

Lemma 1.2**.**

$H$* is Raynaud if and only if $M(H)$ is balanced.*

Proof.

We may assume that $k$ is algebraically closed, as both conditions are invariant under passage to an algebraic closure. If $H$ is étale, it is constant, and must then be isomorphic, with the $\mathcal{O}_{E}$ action, to $\mathcal{O}_{E}\otimes\mathbb{Z}/p\mathbb{Z},$ whose Dieudonné module is evidently balanced. Similarly, if $H$ is multiplicative.

There remains the local–local case. As a *scheme, *stripped of the group structure, $H$ is then either (i) $Spec(k[X]/(X^{p^{2}}))$ or (ii) $Spec(k[X,Y]/(X^{p},Y^{p})),$ where the second case occurs if and only if $H$ is killed by the Frobenius morphism ${\rm Fr}\colon H\to H^{(p)}$. Since $I$ is of codimension 1 and, in the local case, also nilpotent, it coincides with the maximal ideal of $A=\mathcal{O}(H).$ The cotangent space at the origin, $I/I^{2}$, is then $k\overline{X}$ in case (i) and $k\overline{X}\oplus k\overline{Y}$ in case (ii).

In case (i) $\kappa$ may act on the one-dimesional $I/I^{2}$ by $\Sigma$ or $\overline{\Sigma}$, and so does the group $\kappa^{\times}$ act. Either way, $\kappa^{\times}$ acts on $I^{i}/I^{i+1}$ ($1\leq i\leq p^{2}-1)$ via $\Sigma^{i}$ (or $\overline{\Sigma}^{i})$, so every character $\omega^{i}:\kappa^{\times}\to k^{\times}$ must occur in $I$ with multiplicity 1, and $H$ is automatically Raynaud. But in case (i) we also have an exact sequence of finite flat $\mathcal{O}_{E}$-group schemes

[TABLE]

Here $H_{1}=\ker({\rm Fr}\colon H\to H^{(p)})$ is a subgroup scheme of rank $p,$ and $H_{1}^{(p)}$ is its image. It follows that in case (i) $M(H)$ is an extension of $M(H_{1})^{(p)}$ by $M(H_{1}),$ so is automatically balanced.

Case (ii) is the only case where the “balanced” condition may fail. In this case ${\rm Fr}$ annihilates $H$ so $V=0$ on $M=M(H)$ and

[TABLE]

(see [C-C-O], B.3.5.6-3.5.7). We find that $M$ is balanced if and only if ${\rm Lie}(H),$ equivalently its dual $I/I^{2}$, is balanced. If this is the case, i.e. both $\Sigma$ and $\overline{\Sigma}$ occur in $I/I^{2}$, we may choose the variables $X$ and $Y$ so that $\kappa^{\times}$ acts on $X$ via $\omega$ and on $Y$ via $\omega^{p},$ so on $X^{i}Y^{j}$ ($i,j<p,$ not both [math]) it acts via $\omega^{i+jp}$ and every character occurs with multiplicity 1 in $I$. Thus $H$ is Raynaud in this case. If, on the contrary, $I/I^{2}$ is $\kappa^{\times}$-isotypical, we can not have $\dim I^{(i)}=1$ for every $i,$ and $H$ is not Raynaud. ∎

Let $H^{D}$ denote the Cartier dual of $H,$ which is also finite flat of rank $p^{2},$ and endow it by an $\mathcal{O}_{E}$-action $\iota^{D}:\mathcal{O}_{E}\rightarrow{\rm{\rm End}}_{R}(H^{D})$ via the formula

[TABLE]

i.e. for any $R$-algebra $R^{\prime}$ and any $x\in H(R^{\prime}),$ $y\in H^{D}(R^{\prime})$,

[TABLE]

Corollary 1.3**.**

$H$* is Raynaud if and only if $H^{D}$ is Raynaud.*

Proof.

$M(H^{D})$ (identified with the contravariant Dieudonné module of $H$) is the $k$-linear dual of $M(H),$ so one is balanced if and only if the other is. ∎

1.2.2. The moduli problem $(S)$

We now define the three integral models for the Shimura varieties with parahoric level structure at $p$ as moduli schemes for moduli problems of PEL type. It is well known and easy to check that in the generic fiber these moduli problems yield the given Shimura varieties. For the relation with the models defined by Rapoport and Zink, and the representability of the moduli problems, see 1.3 below.

The Picard modular surface $S$ has a smooth integral model $\mathscr{S}$ over $\mathcal{O}_{p}$. It is a fine moduli scheme for the moduli problem which assigns to each $\mathcal{O}_{p}$-algebra $R$ isomorphism classes of tuples $\underline{A}=(A,\phi,\iota,\eta),$ where

• •

$A$ is an abelian $3$-fold over $R$.

• •

$\phi\colon A\overset{\sim}{\to}A^{t}$ is a principal polarization.

• •

$\iota\colon\mathcal{O}_{E}\to{\rm{\rm End}}_{R}(A)$ is a ring homomorphism, such that the Rosati involution induced by $\phi$ on ${\rm{\rm End}}_{R}(A)$ preserves its image, and is given on it by $\iota(a)\mapsto\iota(\overline{a}).$ We furthermore require that ${\rm Lie}(A)$ becomes an $\mathcal{O}_{E}$-module of type $(2,1)$ in the sense that it is the direct sum of a locally free $R$-module of rank 2 on which $\iota(a)_{*}$ acts like the image of $a$ in $R$, and a locally free rank 1 module on which it acts like $\overline{a}.$

• •

$\eta$:$\Lambda/N\Lambda\simeq A[N]$ is a full level-$N$ $\mathcal{O}_{E}$-structure (recall $p\nmid N\geq 3$).

Our reference to moduli problems and representability is the comprehensive volume by Lan. In particular, we refer the reader to the precise definition of level structure given there ([Lan] 1.3.6.2), and to the condition of étale liftability. In addition to being compatible with the $\mathcal{O}_{E}$-action, $\eta$ should carry the polarization pairing

[TABLE]

derived from $(,)$ to the Weil $e_{N}$-pairing induced by $\phi$ on $A[N]\times A[N].$ Part of the data involved in $\eta$ is an isomorphism between the (étale) target groups of the two pairings: $\nu_{N}:\mathbb{Z}/N\mathbb{Z}\simeq\mu_{N}$, making the last condition meaningful. These isomorphisms form a torsor $\underline{\text{Isom}}(\mathbb{Z}/N\mathbb{Z},\mu_{N})$ under $(\mathbb{Z}/N\mathbb{Z})^{\times}$, and in this way $\nu_{N}$ becomes a morphism form $\mathcal{\mathscr{S}}$ to $\underline{\text{Isom}}(\mathbb{Z}/N\mathbb{Z},\mu_{N}),$ regarded as a scheme over $\mathcal{O}_{p}$ of relative dimension 0. We call $\nu_{N}$ the multiplier morphism.

1.2.3. The moduli problem $(\widetilde{S})$

The Shimura variety $\widetilde{S}$ has an integral model $\mathscr{\widetilde{S}}$ over $\mathcal{O}_{p}.$ It is a fine moduli scheme for the moduli problem which assigns to each $\mathcal{O}_{p}$-algebra $R$ isomorphism classes of tuples $\underline{A}^{\prime}=(A^{\prime},\psi,\iota^{\prime},\eta^{\prime}),$ where

• •

$A^{\prime}$ is an abelian $3$-fold over $R$.

• •

$\psi\colon A^{\prime}\to A^{\prime t}$ is a polarization of degree $p^{2}$.

• •

$\iota^{\prime}\colon\mathcal{O}_{E}\to{\rm{\rm End}}_{R}(A^{\prime})$ is a ring homomorphism, satisfying the same requirements as for $(S).$ In addition, we require that $\ker(\psi)$ is preserved by $\iota^{\prime}(\mathcal{O}_{E})$ and is Raynaud.

• •

$\eta^{\prime}$ is a full level-$N$ $\mathcal{O}_{E}$-structure.

1.2.4. The moduli problem $(S_{0}(p))$

The Shimura variety $S_{0}(p)$ has an integral model $\mathscr{S}_{0}(p)$ over $\mathcal{O}_{p}.$ It is a fine moduli scheme for the moduli problem which assigns to each $\mathcal{O}_{p}$-algebra $R$ isomorphism classes of tuples $(\underline{A},H)=(A,\phi,\iota,\eta,H),$ where

• •

$\underline{A}$ is as in $(S)$

• •

$H\subset A[p]$ is a Raynaud $\mathcal{O}_{E}$-subgroup scheme of rank $p^{2},$ which is isotropic for the Weil pairing $e_{p}$ (the Mumford pairing $e_{p\phi}$ attached to the polarization $p\phi).$

1.2.5. The maps between the integral models

There are projection maps

[TABLE]

extending the maps of Proposition 1.1. The map $\pi$ is neither finite, nor flat anymore. On the moduli problem, it is simply “forget $H$”.

The second map $\widetilde{\pi}$ is defined as follows. Pick $(\underline{A},H)\in\mathscr{S}_{0}(p)(R).$ Let $A^{\prime}=A/H.$ Since $H$ is isotropic for $e_{p\phi},$ its annihilator in this pairing is a finite flat subgroup scheme $H\subset H^{\perp}\subset A[p]$ and $A[p]/H^{\perp}\simeq H^{D}.$ We claim that $H^{\perp}/H$ is Raynaud. We may assume that $R=k$ is an algebraically closed field of characteristic $p$. As both $H$ and $H^{D}$ are Raynaud, $M(H)$ and $M(A[p]/H^{\perp})$ are balanced. It follows that $M(H^{\perp}/H)$ is also balanced, so $H^{\perp}/H$ is Raynaud. The polarization $p\phi$ descends canonically to a polarization $\psi:A^{\prime}\to(A^{\prime})^{t}$ whose kernel is $\ker(\psi)=H^{\perp}/H.$ Its degree is $p^{2}.$ Finally $\iota^{\prime}$ and $\eta^{\prime}$ are defined naturally from $\iota$ and $\eta.$ To check that we obtained a point of $\widetilde{\mathscr{S}}$, we need only check one non-trivial222In characteristic 0, or if $H$ is étale, this is obvious, because the Lie algebra is not changed, but in characteristic $p$ the type of the Lie algebra may well change under an isogeny. point, that ${\rm Lie}(A^{\prime})$ is indeed of type $(2,1).$ This can be seen, using the Raynaud condition, as follows. We may assume again that $R=k$ is an algebraically closed field containing $\kappa.$ The exact sequence

[TABLE]

yields, in covariant Dieudonné theory, exact sequences333A guide for the perplexed: the covariant Dieudonné modules of a finite flat group scheme (resp. $p$-divisible group) is defined as the contravariant Dieudonné module of its Cartier (resp. Serre) dual. From the exact sequence $0\to H^{D}\to A^{\prime t}\to A^{t}\to 0$ we get the top row of the diagram. and a commutative diagram

[TABLE]

where we have abbreviated $M(A)=M(A[p^{\infty}])$ etc. The snake lemma yields

[TABLE]

[TABLE]

Thus the type of ${\rm Lie}(A^{\prime})$ is also $(2,1)$ if and only if $M(H)[V]$ and $M(H)^{(p)}/VM(H)$ have the same type. But from the exact sequence

[TABLE]

we see that this is the case if and only if $M(H)$ is balanced. We conclude that $H$ being Raynaud is in fact a necessary and sufficient condition for $A^{\prime}=A/H$ to be of type $(2,1)$ as well.

We shall see later that in contrast to $\pi,$ the map $\widetilde{\pi}$ is finite flat of degree $p+1$.

If we denote by $f:A\to A^{\prime}$ the canonical homomorphism with kernel $H,$ and identify $A^{\prime t}$ with $A/H^{\perp},$ then $f^{\prime t}:A^{\prime t}\to A^{t}$ has kernel $A[p]/H^{\perp}$ and

[TABLE]

1.2.6. The moduli problem $(\widetilde{S}_{0}(p))$

There is a fourth moduli problem that one can define. It turns out to be equivalent to $(S_{0}(p))$, yet useful for later calculations and for the study of the moduli problem $\mathscr{T}$ mentioned in the introduction.

The moduli problem $(\widetilde{S}_{0}(p))$ assigns to every $\mathcal{O}_{p}$-algebra $R$ isomorphism classes of tuples $(\underline{A}^{\prime},J)$ where

• •

$\underline{A}^{\prime}$ is as in $(\widetilde{S}$)

• •

$J\subset A^{\prime}[p]$ is a finite flat $\mathcal{O}_{E}$-subgroup scheme of rank $p^{4},$ containing $\ker(\psi),$ such that $J/\ker(\psi)$ is Raynaud, and which is maximal isotropic for the Mumford pairing $e_{p\psi}$.

Note that $\deg(p\psi)=p^{8}.$

Proposition 1.4**.**

The moduli problems $(S_{0}(p))$ and $(\widetilde{S}_{0}(p))$ are equivalent, hence $(\widetilde{S}_{0}(p))$ is also represented by $\mathscr{S}_{0}(p).$

Proof.

To pass from $(\underline{A},H)$ to $(\underline{A}^{\prime},J)$ define

[TABLE]

and observe that $J/\ker(\psi)=A[p]/H^{\perp}$ is Raynaud, and that $J$ is isotropic (hence, from degree considerations, maximal isotropic) for $e_{p\psi}.$ To pass from $(\underline{A}^{\prime},J)$ to $(\underline{A},H)$ define $A=A^{\prime}/J$, descend $p\psi$ to obtain a principal polarization $\phi$ on $A,$ and let $H=A^{\prime}[p]/J.$ We leave to the reader the verification that we obtain a point of $(\widetilde{S}_{0}(p))$, as well as that these two constructions are inverse to each other. ∎

In terms of this new interpretation of $\mathscr{S}_{0}(p)$ the map $\widetilde{\pi}$ is simply “forget $J$”.

Proposition 1.5**.**

The schemes $\mathscr{S},\mathscr{S}_{0}$(p) and $\mathscr{\widetilde{S}}$ are regular. They are flat over $\mathcal{O}_{p}$ and their special fibers are reduced. The maps $\pi$ and $\widetilde{\pi}$ are surjective and proper.

Proof.

The “flat” and “reduced” assertions follow from the Main Result of [Gö], and from the fact that locally for the étale topology, a neighborhood of a point in the special fiber of $\mathscr{S}_{0}(p)$ or $\mathscr{\widetilde{S}}$ is isomorphic to an open neighborhood in the local model. Similarly, regularity follows from the determination of the completed local rings of the three schemes in [Bel] III.3.4.8. Although Bellaïche does not use the formalism of [Ra-Zi], he builds upon the earlier work of de Jong [dJ2], which except for the notation, yields identical results for the completed local rings as what one would get from the more general theory developed by Rapoport and Zink.

Properness and surjectivity of $\pi$ and $\widetilde{\pi}$ are usually proved along with the proof of the representability of $\mathscr{S}_{0}(p).$ For the map $\pi$ it is done in [Bel] III.3.2.3. For the map $\widetilde{\pi}$ the proof is similar, and we only sketch it. It is best described with the new interpretation of $\mathscr{S}_{0}(p)$ as representing the moduli problem $(\widetilde{S}_{0}(p)).$ Consider first a larger moduli problem $(\widetilde{S}_{0}(p)^{\prime})$ obtained from $(\widetilde{S}_{0}(p))$ by* relaxing* the Raynaud condition on $J/\ker(\psi).$ One proves, following de Jong, that this modified moduli problem is proper and surjective over $(\widetilde{S}).$ Properness follows from the valuative criterion. The Raynaud condition is a closed condition, a fact which secures the properness of $\widetilde{\pi}.$ Surjectivity clearly holds in the generic fiber. By [Gö], the generic fiber of $\mathscr{\widetilde{S}}$ is dense. Since $\widetilde{\pi}$ is already known to be proper, its image must be closed, hence is everything. ∎

1.2.7. Diamond operators

If $a\in(\mathcal{O}_{E}/N\mathcal{O}_{E})^{\times}$ we denote by $\left\langle a\right\rangle$ the automorphism of $\mathscr{S}$, defined on the moduli problem by

[TABLE]

The same notation will be applied to the other moduli schemes.

1.3. Translation into the language

of Rapoport and Zink

The moduli problems that we defined in the preceding sections are examples of the moduli problems defined in chapter 6 of [Ra-Zi], although the Raynaud condition is implicit there, as we shall now explain. It follows (from general results of Kottwitz) that, as has been claimed above, they are indeed representable by fine moduli schemes when $N\geq 3$. We remark that [Bel] gives an independent proof of the representability of $(S_{0}(p))$ by proving that it is relatively representable over $(S).$

Using the notation of [Ra-Zi] we take $B=E,\,\mathcal{O}_{B}=\mathcal{O}_{E},\,V=E^{3}$ as before and $b^{*}=\overline{b}.$ Let $\mathcal{L},\mathcal{\widetilde{L}}$ and $\mathcal{L}_{0}(p)$ be the following *self-dual lattice chains *in $V_{p}$ (see (1.1)):

[TABLE]

[TABLE]

[TABLE]

View the three lattice chains as categories, inclusions as morphisms. The moduli problem of type $(\mathcal{L})$, as defined in [Ra-Zi] Definition 6.9, is clearly our $(S);$ just set $A=A_{\Lambda_{0}}.$

The moduli problem of type $(\mathcal{\widetilde{L}})$ is our $(\widetilde{S}).$ Recall the definition of a “principally polarized $\mathcal{\widetilde{L}}$-set of abelian varieties of type $(2,1)$” over a base ring $R$ as above ([Ra-Zi], Definition 6.6). First, one is given the $\mathcal{\widetilde{L}}$-set of abelian schemes $A_{\Lambda_{\bullet}}$ of type $(2,1)$. Then one gives the “principal polarization”444We apologize for the unintentional double meaning attributed to tilde. We chose to denote the moduli problem $(\widetilde{S})$ with a tilde, hence it made sense to denote the corresponding lattice chain also $\mathcal{\widetilde{L}}$. In [Ra-Zi], passing to the dual $\mathcal{\widetilde{L}}$-set is also denoted by a tilde, hence the tilde on $\widetilde{A}_{\Lambda_{\bullet}}$. \lambda:$$A_{\Lambda_{\bullet}}\simeq\widetilde{A}_{\Lambda_{\bullet}}. Note that the $\mathcal{\widetilde{L}}$-set $\widetilde{A}_{\Lambda_{\bullet}}$ is of type $(1,2)$ because $\lambda$ induces the Rosati involution on the endomorphism ring, hence switches types. We set

[TABLE]

Then $\psi$ is a polarization in the ordinary sense, of degree $p^{2}=[\Lambda_{1}:\Lambda_{2}]$. If $R=k$ is an algebraically closed field in characteristic $p,$

[TABLE]

([Ra-Zi] 6.10) is balanced, so $\ker(\psi)$ is Raynaud. Conversely, if we are given data as in $(S)$, thanks to the fact that $\ker(\psi)$ is Raynaud the signature of $A^{\prime\prime}=A^{\prime}/\ker(\psi)$ (with $\mathcal{O}_{E}$-action induced by $\iota^{\prime}$) is $(2,1)$ (as explained at the end of 1.2.4), so we can define

[TABLE]

and “polarize” the resulting $\mathcal{\widetilde{L}}$-set by letting $\lambda$ be the unique type-reversing isomorphism of $\mathcal{\widetilde{L}}$-sets satisfying $\psi=\lambda\circ\rho_{\Lambda_{1},\Lambda_{2}}.$

The proof that the moduli problem of type $(\mathcal{L}_{0}(p))$ is our $(S_{0}(p))$ is in principle identical, and we only sketch it. Once again, given the data $(S_{0}(p))$ we construct an $\mathcal{L}_{0}(p)$-set of abelian varieties by interlacing the previous two constructions. First, letting $A=A_{\Lambda_{0}}\simeq A_{p\Lambda_{0}}$ we use the Raynaud condition on $H$ to ensure that $A^{\prime}=A/H=A_{\Lambda_{2}}$ is of type $(2,1).$ Then we continue and define $A_{\Lambda_{1}}=A/H^{\perp}$ and the polarization $\lambda$ as before.

1.4. Modular curves on the Picard modular surface

1.4.1. Embedding the modular curve

Maps between Shimura data induce maps between Shimura varieties. Here we have unitary groups of signature $(1,1)$ at infinity mapping (in many ways) to our $\boldsymbol{G}.$ These group homomorphisms give rise to morphisms of modular curves and Shimura curves to our Picard modular surface. Rather than go through the familiar yoga of Shimura data, we jump straight ahead to the moduli interpretation, thereby giving the morphism on the level of integral structures. We give only one example, which will be explored in connection with the geometry of the special fiber at $p$ later on.

Let $B_{0}$ be a fixed elliptic curve defined over $\mathcal{O}_{p}$ with complex multiplication by $\mathcal{O}_{E}$ and CM type $\Sigma$. Such a curve exists because $(p)$ splits completely in the Hilbert class field $H$ of $E$, and if $\mathfrak{P}$ is a prime divisor of $(p)$ in $H,$ $B_{0}$ may be defined over $\mathcal{O}_{H,\mathfrak{P}}=\mathcal{O}_{p}$. The reduction of $B_{0}$ modulo $p$ is a supersingular elliptic curve defined over $\kappa$. Let $\phi_{0}:B_{0}\simeq B_{0}^{t}$ be the canonical principal polarization of $B_{0}$, and $\iota_{0}:\mathcal{O}_{E}\simeq{\rm{\rm End}}(B_{0})$.

Recall that $p\nmid N\geq 3.$ Let $-D$ be the discriminant of $E$ and $\delta=\sqrt{-D}$ a fixed square root of it in $E$. Assume for simplicity that $D$ is odd and $(N,D)=1$ (otherwise the construction below has to be modified slightly). Let $\mathscr{Z}_{0}$ be the scheme parametrizing $\mathcal{O}_{E}$-isomorphisms $\eta_{0}:\mathcal{O}_{E}/N\mathcal{O}_{E}\simeq B_{0}[N]$. It is étale of relative dimension [math] over $\mathcal{O}_{p}$ and comes with a “multiplier morphism” $\nu_{N}$ to $\underline{\text{Isom}}(\mathbb{Z}/N\mathbb{Z},\mu_{N})$. Write

[TABLE]

for an $R$-valued point of $\mathscr{Z}_{0}$.

Let $\mathscr{X}=X_{0}(D;N)$ be the modular curve parametrizing elliptic curves $B$ with a full level $N$ structure $\nu:(\mathbb{Z}/N\mathbb{Z})^{2}\simeq B[N]$ and a cyclic subgroup scheme $M$ of order $D.$ We view $\mathscr{X}$ as a scheme over $\mathcal{O}_{p}.$ It too comes equipped with a “multiplier morpism” $\nu_{N}$ to $\underline{\text{Isom}}(\mathbb{Z}/N\mathbb{Z},\mu_{N}).$ If we identify $\det(B[N])$ with $\mu_{N}$ via the Weil pairing, then $\nu_{N}=\det\nu$. We remark that $\mathscr{X}$ is neither complete (the cusps are missing) nor connected ($\det\nu$ is not fixed), and that every subgroup scheme $M$ as above is étale, since $D$ is invertible.

Let $R$ be an $\mathcal{O}_{p}$-algebra and $\underline{B}=(B,\nu,M)\in\mathscr{X}(R).$ Let $A_{1}(\underline{B})$ be the abelian surface $\mathcal{O}_{E}\otimes B/(\delta\otimes M)$. As $D$ is odd, hence square-free, every class in $\mathcal{O}_{E}/\delta\mathcal{O}_{E}$ is represented by a rational integer. As $\delta$ kills $\delta\otimes M$, this subgroup is $\mathcal{O}_{E}$-stable. It is also maximal isotropic for the Mumford pairing induced by the canonical degree $D^{2}$ polarization

[TABLE]

The identification $\delta^{-1}\mathcal{O}_{E}\otimes B=(\mathcal{O}_{E}\otimes B)^{t}$ is such that the resulting Weil $e_{n}$-pairing between $\mathcal{O}_{E}\otimes B[n]$ and $\delta^{-1}\mathcal{O}_{E}\otimes B[n]$ is

[TABLE]

where $e_{n}^{B}$ is Weil’s $e_{n}$-pairing on $B[n].$ We may therefore descend $\phi_{1}^{\prime}$ to obtain a principal polarization $\phi_{1}$ of $A_{1}(\underline{B})$. We let $\iota_{1}$ be the natural action of $\mathcal{O}_{E}$ as endomorphisms of $A_{1}(\underline{B})$. It is of type $(\Sigma,\overline{\Sigma}).$ Let

[TABLE]

a full level-$N$ $\mathcal{O}_{E}$-structure.

Let $\underline{B}_{0}\in\mathscr{Z}_{0}(R)$ and $\underline{B}\in\mathscr{X}(R)$ be such that $\nu_{N}(\underline{B}_{0})=\nu_{N}(\underline{B}).$ Define

[TABLE]

The structure $\underline{A}(\underline{B}_{0},\underline{B})=(A,\phi,\iota,\eta)\in\mathscr{S}(R).$ Indeed, the assumption $\nu_{N}(\underline{B}_{0})=\nu_{N}(\underline{B})$ allows us to define a multiplier for $\eta$ so that it becomes compatible with $\phi$, and the rest is obvious. This construction depends functorially on the input. In this way we have defined a morphism

[TABLE]

A minor modification of this construction yields a morphism

[TABLE]

when we add a cyclic subgroup of order $p$ to the level.

1.4.2. Endomorphism rings of $\mathbb{\overline{F}}_{p}$

points of $\mathscr{S}$

Let $D$ be an indefinite quaternion algebra over $\mathbb{Q}$ equipped with a positive involution $\dagger$ and assume that $E$ embeds in $D$ as a $\dagger$-stable subfield. Then

[TABLE]

where $\xi^{2}>0$ is rational, $\xi a\xi^{-1}=\overline{a}$ for $a\in E$, $a^{\dagger}=\overline{a}$ and $\xi^{\dagger}=\xi.$ Furthermore $E$ is the unique quadratic imaginary $\dagger$-stable subfield of $D.$ Let $\mathcal{O}_{D}$ be a maximal order in $D$ such that $\mathcal{O}_{D}\cap E=\mathcal{O}_{E}.$ In this situation we may define the Shimura curve $\mathscr{X}_{D}$ parametrizing abelian surfaces $A_{1}$ with endomorphisms by $\mathcal{O}_{D},$ a principal polarization inducing $\dagger$ as the Rosati involution on $D$, and a full level $N$ structure. Precisely as for the modular curve, we get a morphism from $\mathscr{Z}_{0}\times_{\underline{\text{Isom}}(\mathbb{Z}/N\mathbb{Z},\mu_{N})}\mathscr{X}_{D}$ to $\mathscr{S}$. Its image in $\mathscr{S}$ is called an embedded Shimura curve.

The points of $\mathscr{S}(\mathbb{\overline{F}}_{p})$ lying on the embedded Shimura curves all represent non-simple abelian varieties. There are, however, points $\underline{A}\in\mathscr{S}(\mathbb{\overline{F}}_{p})$ for which $A$ is simple. We use the Honda-Tate theorem to construct them. More precisely, we construct $A$’s with ${\rm{\rm End}}^{0}(A)=M$ a CM field of degree 6.

Let $L$ be a totally real non-Galois cubic field, in which $p$ decomposes as $\mathfrak{p}\mathfrak{q},$ where $f(\mathfrak{p}/p)=2$ and $f(\mathfrak{q}/p)=1.$ Then $M=LE$ is a degree $6$ CM field and $\mathfrak{p}=P\overline{P}$ splits in $M$, while $\mathfrak{q}=Q$ remains inert. Let $\pi$ be an element of $M$ such that $(\pi)=P^{2h}Q^{h}$, where $h$ kills the class of $P^{2}Q$ in the class group of $M$. Then $\pi\overline{\pi}=\epsilon p^{2h}$ for a unit $\epsilon$ of $L$. Replacing $\pi$ with $\epsilon^{-1}\pi^{2}$ and $h$ with $2h$ we may assume that $\epsilon=1.$

Let $q=p^{2h}.$ Then $\pi$ is a Weil $q$-number, and the Honda-Tate theorem implies that there exists a simple $3$-dimensional abelian variety over $\mathbb{F}_{q}$ with ${\rm{\rm End}}(A)$ equal to an order of $M$, and whose Frobenius of degree $q$ is $\pi$. It is easily seen that $A$ is absolutely simple. Changing $A$ by an isogeny if necessary we may assume that ${\rm{\rm End}}(A)\mathcal{\supset O}_{E}$, and that $A$ carries a principal polarization. Of course, $End^{0}(A)=M.$

Since ${\rm{\rm End}}^{0}(A)$, for any $\underline{A}\in\mathscr{S}(\mathbb{\overline{F}}_{p})$, must contain a $6$-dimensional semi-simple $\mathbb{Q}$-algebra, we see that the “most general” $\overline{\mathbb{F}}_{p}$-point of $\mathscr{S}$ carries an abelian variety with CM by a field of degree 6. Generic points of the special fiber of $\mathscr{S}$, by contrast, have no endomorphisms except for $\iota(\mathcal{O}_{E}).$

2. The structure of the special fiber of $\mathscr{S}$

2.1. Stratification

Let $k$ be a fixed algebraic closure of $\kappa.$ Since we shall have no use for the generic fibers of our integral models any more, we denote from now on by $S,$ $\widetilde{S}$ and $S_{0}(p)$ their geometric special fibers, which are schemes defined over $k.$ We denote by $\mathcal{A}$ the universal abelian scheme over $\mathscr{S}$, and by $\mathcal{A}_{x}$ its fiber over a geometric point $x\in S(k).$

Let $\mathfrak{G}$ be the unique (up to isomorphism) connected 1-dimensional $p$-divisible group over $k$ of height $2$. It is self-dual of slope $1/2,$ and isomorphic to the $p$-divisible group of any supersingular elliptic curve over $k$. Fix an embedding $\lambda:\mathcal{O}_{p}\hookrightarrow{\rm{\rm End}}_{k}(\mathfrak{G})$ in which $a\in\mathcal{O}_{p}$ acts on ${\rm Lie}(\mathfrak{G})$ via the natural homomorphism $\Sigma:\mathcal{O}_{p}\twoheadrightarrow\kappa\hookrightarrow k,$ and denote the pair $(\mathfrak{G},\lambda)$ by $\mathfrak{G}_{\Sigma}.$ Let $\mathfrak{G}_{\overline{\Sigma}}$ be the same $p$-divisible group with the embedding $\lambda\circ\sigma$, under which the action of $a\in\mathcal{O}_{p}$ on ${\rm Lie}(\mathfrak{G})$ is via $\overline{\Sigma}=\Sigma\circ\sigma.$

The following theorem is due to Vollaard [Vo], in particular §6. See also [dS-G1] Theorem 2.1.

Theorem 2.1**.**

(i) The special fiber $S$ of $\mathscr{\mathscr{S}}$ is the union of 3 locally closed strata defined over $\kappa$. The $\mu$-ordinary* stratum $S_{\mu}$ is open and dense, and $x\in S_{\mu}(k)$ if and only if*

[TABLE]

as $p$-divisible groups with $\mathcal{O}_{E}$-action. Its complement, $S-S_{\mu}=S_{ss}$ is called the supersingular locus. It is a reduced (but reducible) complete curve, and if $x\in S_{ss}(k)$ then $\mathcal{A}_{x}[p^{\infty}]$ is supersingular, i.e. its Newton polygon is of constant slope $1/2.$ The superspecial locus $S_{ssp}\subset S_{ss}$ is [math]-dimensional and a point $x\in S_{ssp}(k)$ if and only if

[TABLE]

as $p$-divisible groups with $\mathcal{O}_{E}$-action. We let $S_{gss}=S_{ss}-S_{ssp}$ and call it the general supersingular locus.

Oort’s $a$-number

[TABLE]

is $1$ if $x\in S_{\mu}(k)$ or $x\in S_{gss}(k)$ and $3$ if $x\in S_{ssp}(k).$ Let $\alpha_{p}(\mathcal{A}_{x})$ be the maximal $\alpha_{p}$-subgroup of $\mathcal{A}_{x}[p].$ The action of $\kappa$ on ${\rm Lie}(\alpha_{p}(\mathcal{A}_{x}))$* has signature $\Sigma$ in the first two cases, and $(\Sigma,\Sigma,\overline{\Sigma})$ in the third case.*

(ii) If $S^{\prime}$ is a connected component of $S$ then $S^{\prime}\cap S_{ss}$ is a connected component of $S_{ss}$. The non-singular locus of $S_{ss}$ is precisely $S_{gss}.$ The irreducible components of $S_{ss}$ are Fermat curves, whose normalizations are isomorphic to the curve

[TABLE]

(iii) If $N\geq N_{0}(p)$ (an integer depending on $p$) the following also holds. The irreducible components of $S_{ss}$ are already non-singular, and isomorphic to $\mathscr{C}.$ Any two of them intersect at most at one point, and if they intersect, this point belongs to $S_{ssp}(k)$ and the intersection is transversal. There are $p^{3}+1$ superspecial points on each irreducible component of $S_{ss}$, and there are $p+1$ irreducible components of $S_{ss}$ intersecting transversally at each $x\in S_{ssp}(k).$

Let $X$ be the geometric special fiber of the modular curve $\mathscr{X}$ which was constructed555We abuse notation and call the curve $\mathscr{Z}_{0}\times_{\underline{\text{Isom}}(\mathbb{Z}/N\mathbb{Z},\mu_{N})}\mathscr{X}$ simply $\mathscr{X}.$ in §1.4. It is a non-singular curve in $S.$ The following corollary is clear from the description of the strata of $S$.

Corollary 2.2**.**

The curve $X$ does not intersect $S_{gss}$. If $\underline{B}\in X(k)$ is such that $\underline{A}(\underline{B})\in S_{ssp}(k)$ then $B$ is supersingular, and vice versa.

2.2. The tangent bundle of $S$

2.2.1. The special line sub-bundle $TS^{+}$

Outside $S_{ssp},$ one may define a natural line sub-bundle $TS^{+}$ of the tangent bundle $TS$ of $S.$ For this recall the following facts from [dS-G1]. Let $\Omega_{\mathcal{A}/S}$ be the sheaf of relative differentials of the universal abelian variety $\mathcal{A}$, and $\omega_{\mathcal{A}}=f_{*}\Omega_{\mathcal{A}/S}$ where $f:\mathcal{A}\to S$ is the structure morphism. Then $\omega_{\mathcal{A}}$ is a rank 3 vector bundle on $S,$ can be identified with the cotangent space of $\mathcal{A}$ at the origin, and admits a decomposition

[TABLE]

into a plane bundle $\mathcal{P}$ on which $\mathcal{O}_{E}$ acts via $\Sigma$ and a line bundle $\mathcal{L}$ on which it acts via $\overline{\Sigma}.$ Let $\Phi\colon S\to S$ be the absolute Frobenius morphism of degree $p,$ and $\mathcal{A}^{(p)}=S\times_{\Phi,S}\mathcal{A}$ the base change of $\mathcal{A}.$ Similar notation will be employed for the base change of the vector bundles $\mathcal{P}$ or $\mathcal{L}.$ The Verschiebung homomorphism {\rm Ver}_{\mathcal{A}/S}\colon$$\mathcal{A}^{(p)}\to\mathcal{A} induces maps

[TABLE]

which, outside $S_{ssp}$, are both of rank 1. At the superspecial points these maps vanish. Let

[TABLE]

Outside the superspecial points, $\mathcal{P}_{0}$ is a line sub-bundle of $\mathcal{P}$. Outside $S_{ss}$, the lines $\mathcal{P}_{0}^{(p)}$ and $V_{\mathcal{L}}(\mathcal{L})$ are distinct, but along $S_{gss}$ they coincide. In fact,

[TABLE]

which is a global section of $\mathcal{L}^{p^{2}-1}$, is the Hasse invariant (cf. [G-N, Appendix B]; one of the main contributions of [G-N] is the construction of the Hasse invariant for unitary Shimura varieties over totally real fields, which is substantially more difficult), and $V_{\mathcal{P}}^{(p)}\circ V_{\mathcal{L}}=0$ is the equation defining $S_{ss}$ as a subscheme of $S.$

The Kodaira-Spencer isomorphism is an isomorphism

[TABLE]

Definition**.**

Outside $S_{ssp},$ we define $TS^{+}$ to be the annihilator of the line bundle ${\rm KS}(\mathcal{P}_{0}\otimes\mathcal{L}).$ We call $TS^{+}$ the special sub-bundle of $TS.$ By an integral curve of $TS^{+}$ we mean a nonsingular curve $C\subset S-S_{ssp}$ for which $TS^{+}|_{C}=TC$, i.e. $TS^{+}$ is tangent to $C$.

Theorem 2.3**.**

(i) $S_{gss}$ is an integral curve of $TS^{+}.$

(ii) The modular curve $X_{ord}=X\cap S_{\mu}$ is an integral curve of $TS^{+}.$

Proof.

Part (i), although not stated there in this form, was proved in [dS-G2] Proposition 3.11. For (ii) observe that if $x\in X_{ord}(k)\subset S_{\mu}(k)$ then we have the decomposition $\mathcal{A}_{x}=B_{0}\times\mathcal{A}_{1,x}$ where $\mathcal{A}_{1}$ is the abelian surface constructed along $X$ from the universal elliptic curve $\mathcal{B}$ (and the universal cyclic subgroup of rank $D)$ as in §1.4. For the cotangent space we have accordingly

[TABLE]

where the first summand is of type $\Sigma$ and the second of type $(\Sigma,\overline{\Sigma}).$ Thus

[TABLE]

As $\mathcal{A}_{1,x}$ is ordinary, $V$ is injective on $\omega_{\mathcal{A}_{1,x}}(\Sigma)$ and

[TABLE]

As $B_{0}$ is constant along $X$, ${\rm KS}(\mathcal{P}_{0}\otimes\mathcal{L}|_{x})\subset\Omega_{S/k}|_{x}$ annihilates the line $T_{x}X\subset T_{x}S.$ This proves that $T_{x}X=TS^{+}|_{x}$ as claimed. ∎

There are many modular curves and Shimura curves like $X$ on $S$, and by similar arguments they are all integral curves of the special sub-bundle. It would be interesting to know if these are the only integral curves of $TS^{+}$ in $S_{\mu}.$ This is an “André-Oort type” question. It would imply, in particular, that there are no integral curves passing through the CM points constructed in §1.4.2. Note that in characteristic $p$ there could be many integral curves tangent to a perfectly nice vector field. The curves $x-c+\lambda y^{p}=0$, for varying $c$ and $\lambda$, are all tangent to the vector field $\partial/\partial y$ in $\mathbb{A}^{2}$, and infinitely many of them pass through any given point. The correct formulation of the problem should probably ask for curves annihilated by a larger class of differential operators. Such a class should contain, besides the differential operators generated by $TS^{+}$, also “divided powers”.

2.2.2. A characterization in terms of generalized Serre-Tate coordinates

We shall now give a second characterization of $TS^{+}$, which relates it to Moonen’s work on generalized Serre-Tate coordinates in $S_{\mu}$. For the following proposition see [Mo], Example 3.3.2 and 3.3.3(d) (case AU, $r=3,$ $m=1$).

Proposition 2.4**.**

Let $x\in S_{\mu}.$ Let $\mathfrak{\widehat{G}}$ be the formal group over $k$ associated with the $p$-divisible group $\mathfrak{G}$ and let $\mathbb{\widehat{G}}_{m}$ be the formal multiplicative group over $k.$ Then the formal neighborhood $Spf(\widehat{\mathcal{O}}_{S,x})$ of $x$ has a natural structure of a $\mathbb{\widehat{G}}_{m}$-torsor over $\mathfrak{\widehat{G}}$. In particular, it contains a canonical copy of $\mathbb{\widehat{G}}_{m}$ sitting over the origin of $\mathfrak{\widehat{G}}$.

Theorem 2.5**.**

Let $x\in S_{\mu}$. Then the line $TS^{+}|_{x}$ is tangent to the canonical copy of $\mathbb{\widehat{G}}_{m}$ in $Spf(\widehat{\mathcal{O}}_{S,x})$.

At a point $x$ lying on a modular curve $X$ as above, the canonical copy of $\mathbb{\widehat{G}}_{m}$ is identified with the classical Serre-Tate coordinate on $X$, i.e. the formal completion of $X$ at $x$ coincides with $i(\mathbb{\widehat{G}}_{m})$ as a closed formal subscheme of $Spf(\widehat{\mathcal{O}}_{S,x}).$ In this case the theorem is a consequence of Theorem 2.3(ii). Our claim can therefore be viewed as an extension of Theorem 2.3(ii) to a general $\mu$-ordinary point, at which the formal curve $i(\mathbb{\widehat{G}}_{m})$ may no longer be “integrated”.

Proof.

Write $\mathbb{\widehat{G}}_{m}=Spf(k[[T-1]])$ with comultiplication $T\mapsto T\otimes T$, and let $i:\mathbb{\widehat{G}}_{m}\hookrightarrow Spf(\widehat{\mathcal{O}}_{S,x})$ be the embedding of formal schemes given by Proposition 2.4. It sends the closed point $1$ of $\mathbb{\widehat{G}}_{m}$ to $x.$ Let $i_{*}$ be the induced map on tangent spaces

[TABLE]

We have to show that $i_{*}(\partial/\partial T)$ annihilates ${\rm KS}(\mathcal{P}_{0}\otimes\mathcal{L})|_{x}.$ This is equivalent to saying that when we consider the pull back $i^{*}\mathcal{A}$ of the universal abelian scheme to $\mathbb{\widehat{G}}_{m}$, its Kodaira-Spencer map kills $\mathcal{P}_{0}\otimes\mathcal{L}|_{1}$. For this recall the definition of ${\rm KS}={\rm KS}(\Sigma)$ from [dS-G1], §1.4.2.

Let $\mathfrak{S}=\mathbb{\widehat{G}}_{m}$ and write for simplicity $\mathcal{A}$ for $i^{*}\mathcal{A}$. We then have the following commutative diagram

[TABLE]

in which we identified $H^{1}(\mathcal{A},\mathcal{O})$ with $H^{0}(\mathcal{A}^{t},\Omega^{1})^{\vee}$ and used the polarization to identify the latter with $\omega_{\mathcal{A}/\mathfrak{S}}^{\vee}$, reversing types. Here $\nabla$ is the Gauss-Manin connection, and the tensor product is over $\mathcal{\widehat{O}_{\mathfrak{S}}}=k[[T-1]].$ Although $\nabla$ is a derivation, ${\rm KS}$ is a homomorphism of vector bundles over $\mathcal{\widehat{O}_{\mathfrak{S}}}$. We shall show that ${\rm KS}(\mathcal{P}_{0})=0,$ where $\mathcal{P}_{0}=\ker(V:\omega_{\mathcal{A}/\mathfrak{S}}\to\omega_{\mathcal{A}/\mathfrak{S}}^{(p)})\cap\mathcal{P}$.

At this point recall the filtration

[TABLE]

of the $p$-divisible group of $\mathcal{A}$ over $\mathfrak{S}.$ The graded pieces are of height 2 and $\mathcal{O}_{E}$-stable. They are rigid (do not admit non-trivial deformations as $p$-divisible groups with $\mathcal{O}_{E}$ action) and given by

[TABLE]

For any $p$-divisible group $G$ over $\mathfrak{S}$ denote by $\mathbb{D}(G)$ the Dieudonné crystal associated to $G$, and let $D(G)=\mathbb{D}(G)_{\mathfrak{S}},$ cf. [Gro]. The $\mathcal{\widehat{O}_{\mathfrak{S}}}$-module $D(G)$ is endowed with an integrable connection $\nabla$ and the pair $(D(G),\nabla)$ determines $\mathbb{D}(G).$

In our case, we can identify $D(\mathcal{A}[p^{\infty}])$ with $H_{dR}^{1}(\mathcal{A}/\mathfrak{S})$, and the connection with the Gauss-Manin connection. The above filtration on $\mathcal{A}[p^{\infty}]$ induces therefore a filtration $Fil^{\bullet}$ on $H_{dR}^{1}(\mathcal{A}/\mathfrak{S})$ which is preserved by $\nabla$. Since the functor $\mathbb{D}$ is contravariant, we write the filtration as

[TABLE]

where

[TABLE]

For example, $Fil^{1}H_{dR}^{1}(\mathcal{A}/\mathfrak{S})$ is sometimes referred to as the “unit root subspace”. As $Fil^{2}\mathcal{A}[p^{\infty}]$ is of multiplicative type, $\ker(V:H_{dR}^{1}(\mathcal{A}/\mathfrak{S})\to H_{dR}^{1}(\mathcal{A}/\mathfrak{S})^{(p)})$ is contained in $Fil^{2}H_{dR}^{1}(\mathcal{A}/\mathfrak{S})$. In particular,

[TABLE]

Let $G=\mathcal{A}[p^{\infty}]/\mathcal{A}[p^{\infty}]^{m}$, so that $Fil^{2}H_{dR}^{1}(\mathcal{A}/\mathfrak{S})=D(G).$ It follows that in computing ${\rm KS}$ on $\mathcal{P}_{0}$ we may use the following diagram instead of (2.1):

[TABLE]

Finally, we have to use the description of the formal neighborhood of $x$ as given in [Mo]. Since we are considering the pull-back of $\mathcal{A}$ to $\mathfrak{S}$ only, and not the full deformation over $Spf(\widehat{\mathcal{O}}_{S,x})$, the $p$-divisible groups $Fil^{1}\mathcal{A}[p^{\infty}]$, and dually $G=\mathcal{A}[p^{\infty}]/Fil^{2}$, are constant over $\mathfrak{S}$. Thus over $\mathfrak{S}$

[TABLE]

and $\nabla$ maps $D(\mathfrak{G})$ to $D(\mathfrak{G})\otimes\Omega_{\mathfrak{S}}^{1}$. Since

[TABLE]

as subspaces of $H_{dR}^{1}(\mathcal{A}/\mathfrak{S}),$

[TABLE]

The bottom arrow in (2.2) comes from the homomorphism

[TABLE]

But the projection $pr$ kills $\mathcal{P}_{0}\subset\omega_{\mathcal{A}/\mathfrak{S}}.$ This concludes the proof. ∎

We shall later show that the line sub-bundle $TS^{+}$ has a third characterization, in connection with the ramification in the covering $\pi\colon S_{0}(p)\to S.$ The definitions and the discussion of this section have obvious generalizations to higher dimensional unitary Shimura varieties. We intend to address them in a future work.

2.3. The blow up of $S$ at the superspecial points

We denote by $S^{\#}$ the surface over $k$ which is obtained by blowing up the superspecial points on $S$. The fiber of $S^{\#}\to S$ above a superspecial point $x$ is a projective line which we denote by $E_{x}$. It is canonically identified with $\mathbb{P}(T_{x}S).$

Since $S$ has a canonical model over $\kappa$ and the stratum $S_{ssp}$ is defined over $\kappa,$ $S^{\#}$ too has a canonical model over $\kappa.$ In fact, it is the fine moduli space of a moduli problem $(S^{\#})$ which is unique to characteristic $p.$ For any $\kappa$-algebra $R$, $S^{\#}(R)$ classifies isomorphism classes of pairs $(\underline{A},\mathcal{P}_{0})$ where

• •

$\underline{A}\in S(R)$

• •

$\mathcal{P}_{0}\subset\ker(V:\omega_{A/R}(\Sigma)\to\omega_{A/R}^{(p)}(\Sigma))$ is a line sub-bundle of $\mathcal{P}=\omega_{A/R}(\Sigma$) which is annihilated by $V$.

If no geometric fiber of $A/R$ is superspecial then $\mathcal{P}_{0}$ is unique. At superspecial points, however, $V$ kills $\mathcal{P},$ so the additional data amounts to a choice of a line in the plane $\mathcal{P}.$

If $N=1$ then $S$ is a stack defined over $\kappa$ and the superspecial points are $\kappa$-rational. It follows that $\mathcal{P}$ is defined over $\kappa$ too and we can equip each

[TABLE]

with a canonical $\kappa$-rational structure. If $N>1$ then level structure at $N$ forces superspecial points to be defined over larger finite fields, but since $\mathcal{P}$ is independent of this extra level structure, the tangent space and the exceptional divisor $E_{x}$ still carry a canonical $\kappa$ structure.

In practice we use a coordinate $\zeta$ on $E_{x}$ which is derived from a particular choice of a basis for the Dieudonné module of $A_{x}$ at $x\in S_{ssp}.$ This will be explained in Theorem 4.11 below.

3. Local structure of the three integral models

3.1. Raynaud’s classification

Recall that $k$ is our fixed algebraically closed field containing $\kappa.$ In [Ray] Raynaud classifies the finite flat group schemes of rank $p^{2}$ over $k,$ which admit an action of $\kappa$ and satisfy the Raynaud condition discussed in 1.2.1. See also [Bel], III.2.3. They are given in the following table.

[TABLE]

Explanations

• •

Each group scheme is designated by a vector $(a_{0},b_{0};a_{1},b_{1})$ with entries from $\{0,1\}$ where $a_{0}b_{0}=a_{1}b_{1}=0.$ There are 9 possibilities. As a scheme $H=Spec(A)$ where $A=k[X,Y]/(X^{p}-b_{0}Y,Y^{p}-b_{1}X).$ The group structure (Hopf algebra structure on $A$) involves the $a_{i}.$ It is completely determined by the condition that the Cartier dual $H^{D}$ is obtained by interchanging $a_{0}$ with $b_{0}$, $a_{1}$ with $b_{1}.$ The twist $\kappa\otimes_{\sigma,\kappa}H$ of $H$ is obtained by interchanging $a_{0}$ with $a_{1},$ and likewise $b_{0}$ with $b_{1}.$

• •

The column ${\rm Lie}(H)$ gives the signature of $\kappa$ on ${\rm Lie}(H),$ with multiplicities.

• •

The column ${\rm Lie}(\alpha_{p}(H))$ gives the signature of $\kappa$, with multiplicities, on the Lie algebra of the maximal $\alpha_{p}$-subgroup of $H$ (whose dimension is Oort’s $a$-number).

• •

The invariants $\alpha,\beta$ are defined by

[TABLE]

They satisfy

[TABLE]

The third invariant, $\gamma,$ is not an intrinsic invariant of $H,$ but rather of the way it sits as an isotropic subgroup of $A[p].$ Recall that if $(\underline{A},H)$ is a point of $S_{0}(p)(k),$ we have a filtration

[TABLE]

with graded pieces $A[p]/H^{\perp}\simeq H^{D}$ and $H^{\perp}/H\simeq\ker\psi$ (see §1.2.5). We then set $\gamma=\dim_{k}{\rm Lie}(H^{\perp}/H)$.

• •

Finally, the last column indicates over which of the strata of $S$ such points $(\underline{A},H)$ lie. A hyphen indicates that an $H$ of the given type does not occur as an isotropic subgroup of $A[p]$ for $\underline{A}$ as in $(S).$ This is the contents of the next lemma.

Lemma 3.1**.**

The subgroups $\mathfrak{G}[p]_{\overline{\Sigma}},$ $\alpha_{p^{2},\overline{\Sigma}}$ and $\alpha_{p^{2},\overline{\Sigma}}^{*}$ do not occur as isotropic subgroups of $A[p]$ for any $\underline{A}$ as in $(S).$

Proof.

We do the first example first. Let $M=M(A[p])$ be the covariant Dieudonné module of $A[p].$ It is a $6$-dimensional vector space over $k,$ with a $\kappa$ action of signature $(3,3),$ and maps $F:M^{(p)}\to M$ and $V:M\to M^{(p)}.$ The principal polarization $\phi$ induces a non-degenerate alternating bilinear pairing $\left\langle,\right\rangle=\left\langle,\right\rangle_{M}:M\times M\to k$ satisfying, for $a\in\kappa,\,\,x\in M^{(p)},\,\,y,u,v\in M$

[TABLE]

[TABLE]

By $\left\langle,\right\rangle_{M^{(p)}}$ we denote the base change of $\left\langle,\right\rangle_{M}$ to $M^{(p)}=k\otimes_{\sigma,k}M$. The first property shows that $M_{0}$ and $M_{1},$ the $\Sigma$- and $\overline{\Sigma}$-eigenspaces of $\kappa,$ are maximal isotropic spaces for the pairing. The second property shows that

[TABLE]

is another maximal isotropic subspace, which, according to our assumption on the signature of $A,$ intersects $M_{0}$ in a $2$-dimensional space, and $M_{1}$ in a line.

Now let $N=M(H)\subset M$ where $H$ is assumed to be of type $\mathfrak{G}[p]_{\overline{\Sigma}}$ and isotropic. Decompose $N=N_{0}\oplus N_{1}$ according to $\kappa$-type. Then ${\rm Lie}(H)=N_{1}$ is orthogonal to $N_{0}$ (because $N$ is isotropic) but also to ${\rm Lie}(A)_{0}=M_{0}[V]$ (because ${\rm Lie}(H)\subset{\rm Lie}(A)$ and ${\rm Lie}(A)$ is isotropic). Since $N_{0}$ is a line lying outside the two-dimensional ${\rm Lie}(A)_{0},$ we deduce that $N_{1}$ is orthogonal to all of $M_{0}$, contradicting the non-degeneracy of the pairing.

The argument for $H\simeq\alpha_{p^{2},\overline{\Sigma}}$ is the same. To rule out $H\simeq\alpha_{p^{2},\overline{\Sigma}}^{*}$ we need another argument, on the $\alpha_{p}$-subgroup. ${\rm Lie}(H)$ alone does not distinguish it from $\alpha_{p^{2},\Sigma}^{*},$ which, as we shall see later, does occur as a possible isotropic subgroup. If $A$ is either $\mu$-ordinary or general supersingular, then the $\alpha_{p}$-subgroup of $A$ is of rank $p$ and type $\Sigma,$ while the $\alpha_{p}$-subgroup of $\alpha_{p^{2},\overline{\Sigma}}^{*}$ is of rank $p$ and type $\overline{\Sigma}.$ Hence, $\alpha_{p^{2},\overline{\Sigma}}^{*}$ is not isomorphic to a subgroup scheme of $A[p].$ If $A$ is superspecial, then its $p$-divisible group is $\mathfrak{G}^{3},$ and does not admit a subgroup scheme of type $\alpha_{p^{2}}^{*}$ at all, because the kernels of Verschiebung and Frobenius on $A^{(p)}$ coincide, while $\alpha_{p^{2}}^{*}$ is killed by Frobenius but not by Verschiebung. ∎

3.2. The completed local rings

3.2.1. Generalities on local models

The method of “local models” was introduced by de Jong [dJ2] and Deligne and Pappas [De-Pa], and developed further by Rapoport and Zink in [Ra-Zi]. See also [P-R-S] and [C-N]. For a point $x$ in the special fiber of a given Shimura variety these authors construct a generalized flag variety, and a point $x^{\prime}$ on it, so that suitable étale neighborhoods of $x$ and $x^{\prime}$ become isomorphic. This allows them to compute the isomorphism type of the completed local rings of the original Shimura variety in terms of linear-algebra data. For the arithmetic schemes $\mathscr{S},$ $\mathscr{S}_{0}(p)$ and $\mathscr{\widetilde{S}}$ these computations were done in [Bel] III.4.3, and in this section we shall quote results from there, adhering as much as possible to the notation used by Bellaïche.

The method of local models is flawed when it comes to functoriality with respect to change of level at $p$. This is because Grothendieck’s theory of the Dieudonné crystal, on which it is based, is functorial in divided power neighborhoods, but not beyond. This flaw appears already in the case of the modular curve $X_{0}(p)$ mapping to the $j$-line $X$. At a supersingular point $y\in X_{0}(p)(k)$ mapping to $x\in X(k)$ we get, for the relation between local models in characteristic $p$

[TABLE]

while the correct model for the pair $\mathcal{\widehat{O}}_{x}\hookrightarrow\mathcal{\widehat{O}}_{y}$ is known to be, ever since Kronecker,

[TABLE]

Observe that modulo $p$th powers of the maximal ideal (where there is a canonical divided power structure) the two models are isomorphic, but over the whole formal neighborhood they are not. The second homomorphism is finite flat of degree $p+1$ while the first is neither finite nor flat.

Despite this flaw, relations between local models of Shimura varieties of PEL type with parahoric level structure suffice to tell us the relations between cotangent spaces, as well as the relations between the infinitesimal deformation theories when we vary the level.

3.2.2. The standard model

Fix $y=[\underline{A},H]\in\mathscr{S}_{0}(p)(k)$. Let $x=\pi(y)\in\mathscr{S}(k)$ and $\widetilde{x}=\widetilde{\pi}(y)\in\widetilde{\mathscr{S}}(k)$. Then $x$ is represented by the tuple $\underline{A}=(A,\phi,\iota,\eta)$ and $\widetilde{x}$ by $\underline{A}^{\prime}=(A^{\prime},\psi,\iota^{\prime},\eta^{\prime})$ where $A^{\prime}=A/H$ and $\psi$ is descended from $p\phi,$ i.e. if $h:A\to A^{\prime}$ is the canonical isogeny with $\ker(h)=H$ then

[TABLE]

Similarly

[TABLE]

Associated with the data $(A,\phi,\iota,A^{\prime},\psi,\iota^{\prime},h)$ is the following linear-algebra data. Let

[TABLE]

be the crystalline Dieudonné modules of the two abelian varieties. Here $\mathbb{D}(A)$ is the (contravariant) Dieudonné crystal associated to $A,$ *cf. *[Gro]. In this section we use crystalline deformation theory as in [Bel]. The translation to covariant Cartier-Dieudonné theory, which will be employed in later sections, is standard (if painful), see the appendix to [C-C-O].

The modules $M_{i}$ are free $W(k)$-modules of rank 6, and decompose under the action of $\mathcal{O}_{E}$ as a direct sum of two rank-3 submodules, denoted $M_{i}(\Sigma)$ and $M_{i}(\overline{\Sigma}).$ The isogeny $h$ induces an injective homomorphism

[TABLE]

respecting the $\mathcal{O}_{E}$-action, whose cokernel is a two-dimensional vector space over $k$ of type $(1,1)$, as $H$ is Raynaud. The polarizations result in type-reversing homomorphism

[TABLE]

where we have used the canonical identifications of $M_{i}^{*}=\text{Hom}(M_{i},W(k))$ with the crystalline Dieudonné modules of the dual abelian varieties. Clearly

[TABLE]

Denote by $\mathcal{M}_{1}$ the coherent sheaf on $\mathscr{S}$ which associates to a Zariski open $U$ the module

[TABLE]

($\mathcal{A}$ being the universal abelian variety over $\mathscr{S})$ and define $\mathcal{M}_{2}$ similarly on $\mathscr{\widetilde{S}}.$ Denote by the same letters their pull-backs to $\mathscr{S}_{0}(p).$ Then the same sort of linear-algebra structure is induced on the sheaves $\mathcal{M}_{i},$ the map $D(h)$ resulting from the canonical isogeny

[TABLE]

where $\mathcal{H}$ is the universal subgroup scheme of $\mathcal{A}$ over $\mathscr{\mathcal{\mathscr{S}}}_{0}(p).$ The following is Théorème III.4.2.5.3 of [Bel].

Theorem 3.2**.**

(i) There exist $W(k)$-bases $\{e_{1},\ldots,e_{6}\}$ of $M_{1}$ and $\{f_{1},\ldots,f_{6}\}$ of $M_{2}$ such that, if we denote by $\{e_{i}^{*}\}$ and $\{f_{i}^{*}\}$ the dual bases, the following properties hold.

(a) $M_{1}(\Sigma)$ is spanned by $\{e_{1},e_{2},e_{3}\},$ $M_{1}(\overline{\Sigma})$ is spanned by $\{e_{4},e_{5},e_{6}\},$ and similarly for $M_{2}.$

(b) The matrices of the homomorphisms $B,B^{\prime}$ in these bases are given by

[TABLE]

i.e. $B(e_{1}^{*})=-e_{6},$ $B^{\prime}(f_{1}^{*})=-pf_{6}$ etc.

(c) The matrix of $D(h)$ is given by

[TABLE]

i.e. $D(h)(f_{1})=e_{1},\,\,D(h)(f_{2})=pe_{2}$ etc.

(ii) The structure $(\mathcal{M}_{1},\mathcal{M}_{2},B,B^{\prime},D(h))$ is locally Zariski isomorphic to

[TABLE]

3.2.3. The Hodge filtration

Fix $y=[\underline{A},H]\in\mathscr{S}_{0}(p)(k)$ as above. The canonical isomorphism

[TABLE]

defines a $3$-dimensional subspace

[TABLE]

which maps isomorphically to $\omega_{A/k},$ and similarly a $3$-dimensional subspace $\omega_{0}^{\prime}\subset M_{2}\otimes_{W(k)}k$ which maps to $\omega_{A^{\prime}/k}.$ These subspaces are $\mathcal{O}_{E}$-invariant of type $(2,1).$ Furthermore, they are isotropic in the sense that if we denote by $\omega_{0}^{\perp}$ the annihilator of $\omega_{0}$ in $M_{1}^{*}\otimes_{W(k)}k$, and similarly for $\omega_{0}^{\prime},$ then

[TABLE]

Equality (rather than inclusion) holds with $B$ because $\phi,$ unlike $\psi,$ is principal. Finally, the map $D(h)$ maps $\omega_{0}^{\prime}$ to $\omega_{0}.$

Lemma 3.3**.**

(i) The invariants $(\alpha,\beta,\gamma)$ at the point $y$ are given by the formulae

[TABLE]

[TABLE]

[TABLE]

(ii) $(\alpha,\beta,\gamma)$ form a complete set of invariants of the structure

[TABLE]

Namely, any two structures (over $k$) of this form having the same set of invariants $(\alpha,\beta,\gamma)$ are isomorphic.

Proof.

Part (ii) is an exercise in linear algebra which we leave out to the reader. In checking it observe that $\alpha$ determines the relative position of $\omega_{0}^{\prime}$ and $\ker D(h),$ $\beta$ determines the relative position of $\omega_{0}$ and $\text{Im}D(h)$, while $\gamma$ is responsible for the relative position of $\ker B^{\prime}$ and $\omega_{0}^{\prime\perp}.$ To prove (i) consider the diagram

[TABLE]

This gives the formulae for $\alpha=\dim_{k}\omega_{H}$ and $\beta=\dim_{k}\omega_{H^{D}}=\dim_{k}\text{coker}(h^{t})^{*\lor}$. The formula for $\gamma$ comes from the fact that if $K=H^{\perp}/H=\ker\psi$ then $\omega_{K}=\omega_{A^{\prime}}/B^{\prime}(\omega_{A^{\prime t}}).$ ∎

3.2.4. Deformations

The following is a consequence of the main theorem of [Gro], characterizing deformations of an abelian variety $A$ (with extra structure) by means of linear-algebra data. See also [dJ2] and [Bel], Proposition III.4.3.6.

Let $\mathcal{C}_{k}$ be the category of local Artinian rings $(R,\mathfrak{m}_{R})$ of residue field isomorphic to $k,$ equipped with an isomorphism $R/\mathfrak{m}_{R}\simeq k$. Observe that every object of $\mathcal{C}_{k}$ comes with a canonical homomorphism $W(k)\to R.$

The local deformation problem $\mathcal{D}$ of the structure $(A,\phi,\iota,A^{\prime},\psi,\iota^{\prime},h)_{/k}$ associates to $R\in\mathcal{C}_{k}$ the set $\mathcal{D}(R)$ of isomorphism classes of similar structures over $R$, equipped with an isomorphism between their reduction modulo $\mathfrak{m}_{R}$ and the given structure over $k$. It is represented by the formal scheme $\text{Spf}(\mathcal{\widehat{O}}_{\mathscr{S}_{0}(p),y}).$ The local model theorem is the following.

Theorem 3.4**.**

The local deformation problem $\mathcal{D}$ is equivalent to the deformation problem $\mathcal{\widetilde{D}}$ which associates to every $(R,\mathfrak{m}_{R})$ as above the set of structures

[TABLE]

satisfying

(a) $\omega$ and $\omega^{\prime}$ are rank-3 direct summands, $\mathcal{O}_{E}$-invariant of type $(2,1),$ reducing modulo $\mathfrak{m}_{R}$ to $\omega_{0}$ and $\omega_{0}^{\prime}$.

(b) $B(\omega^{\perp})=\omega,\,\,\,B^{\prime}(\omega^{\prime\perp})\subset\omega^{\prime}$.

(c) $D(h)(\omega^{\prime})\subset\omega.$

Similar results hold for the moduli problems represented by $\text{{Spf}}(\mathcal{\widehat{O}}_{\mathscr{S},x})$ and $\text{{Spf}}(\mathcal{\widehat{O}}_{\widetilde{\mathscr{S}},\widetilde{x}})$, obtained by forgetting part of the data.

The theorem allows us to compute, quite easily, the complete local rings \mathbf{L}_{y},\,\text{}$$\mathbf{L}_{x} and $\mathbf{L}_{\widetilde{x}}$ representing the deformation problem $\widetilde{\mathcal{D}}$, and deduce isomorphisms

[TABLE]

Since the local deformation problems $\widetilde{\mathcal{D}}$ at $x$ and $\widetilde{x}$ are obtained from the same problem at $y$ by forgetting part of the data, we get canonical homomorphisms

[TABLE]

between the local models. However, as remarked above, this diagram is *not *isomorphic to the corresponding diagram of homomorphisms between the completed local rings of the Picard modular schemes. The best one can get from the general theory is the following.

Theorem 3.5**.**

In the above situation the diagrams $\mathbf{L}_{\widetilde{x}}\to\mathbf{L}_{y}\leftarrow\mathbf{L}_{x}$ and $\mathcal{\widehat{O}}_{\widetilde{\mathscr{S}},\widetilde{x}}\to\mathcal{\widehat{O}}_{\mathscr{S}_{0}(p),y}\leftarrow\mathcal{\widehat{O}}_{\mathscr{S},x}$ become canonically isomorphic after one divides all the local rings by the $p$th powers of their maximal ideals. In particular, they induce isomorphic diagrams on cotangent spaces.

3.3. Computations

3.3.1. Local model diagrams

Let $W=W(k)$ be the ring of Witt vectors of $k$. The scheme $\mathscr{S}$ is smooth over $W,$ so all its completed local rings are isomorphic to $\mathbf{L}_{x}=W[[r,s]].$ In the following table we catalog the diagrams (3.1) giving the local models at $x,\widetilde{x}$ and $y,$ and the maps between them.

Proposition 3.6**.**

For a suitable choice of local parameters the local model diagram is given by the following table (where $\mathbf{L}_{x}=W[[r,s]]$)

[TABLE]

Explanations

• •

The first column indicates the stratum to which $x$ belongs and the possible Raynaud types of the subgroup $H$ in the fiber of $\pi$ above $x.$ The parentheses distinguishing the two cases where $H\simeq\kappa\otimes\alpha_{p}$ refer to the value of the coordinate $\zeta$ on the projective line $F_{x}\subset\pi^{-1}(x)$. This line maps isomorphically to $E_{x}\subset S^{\#}$ and we endow it with the coordinate $\zeta$ as in Section 2.3 and Theorem 4.11 below. The last entry in the table refers to points where $\zeta^{p+1}=-1,$ “generic” refers to all the rest.

• •

The last column refers to the enumeration of the various cases in Bellaïche’s thesis [Bel] III.4.3.8 (cas.sous-cas.sous-sous-cas).

The table implies that the special fiber $S_{0}(p)$ of $\mathscr{\mathscr{S}}_{0}(p)$ is equidimensional of dimension 2. As we shall see in Theorems 4.1 and 4.5, it is the union of three smooth surfaces intersecting transversally. These surfaces are the closures of the strata denoted below by $Y_{m},Y_{et}$ and $Y_{gss}.$ The first two are irreducible, but the third has several connected components. The non-singular points of $S_{0}(p)$, lying on only one of these surfaces, support an $H$ of type $\kappa\otimes\mu_{p},$ $\kappa\otimes\mathbb{Z}/p\mathbb{Z}$ or $\mathfrak{G}[p]$. The points lying on the intersection of two of them support an $H$ of type $\alpha_{p^{2}}^{*},\alpha_{p^{2}}$ or $\kappa\otimes\alpha_{p}$ (generic). The remaining points, represented by the last row in the table, are those where all three surfaces meet.

The special fiber $\widetilde{S}$ of $\widetilde{\mathscr{S}}$ is the union of two smooth surfaces intersecting transversally. One of them, which is the closure of $\widetilde{\pi}(Y_{m})=\widetilde{\pi}(Y_{et})$, is irreducible. The other one, which is the closure of $\widetilde{\pi}(Y_{gss})$, has several connected components. A point $\widetilde{x}=\widetilde{\pi}(y)$ lies on the intersection of these two surfaces if and only if $y$ supports an $H$ of type $\kappa\otimes\alpha_{p}\,\,(\sqrt[p+1]{-1})$, $\alpha_{p^{2}}^{*}$ or $\alpha_{p^{2}}.$

In the next subsections we work out two sample cases from the table, explaining how one arrives at the given description of the local model diagram.

3.3.2. First example

Assume that $x=\pi(y)$ is a gss point and $y\in S_{0}(p)(k)$ is such that $H\simeq\alpha_{p^{2},\Sigma}$ (case I.2 in [Bel]). Here the invariants $(\alpha,\beta,\gamma)=(1,2,2)$. Using Lemma 3.3 one deduces that we may take, without loss of generality,

[TABLE]

A little computation yields that the most general deformation satisfying (a) (b) and (c) of Theorem 3.4 is given by

[TABLE]

[TABLE]

where $r,s,a,b,c\in\mathfrak{m}_{R}$ satisfy the relations

[TABLE]

It follows that

[TABLE]

In the special fiber we get

[TABLE]

where $s\mapsto b$ and $r\mapsto 0.$

Corollary 3.7**.**

The map $\mathcal{\widehat{O}}_{\widetilde{S},\widetilde{x}}\to\mathcal{\widehat{O}}_{S_{0}(p),y}$ is an isomorphism. Identify $\mathcal{\widehat{O}}_{S_{0}(p),y}$ with $\mathbf{L}_{y}\otimes_{W(k)}k.$ There are two analytic branches of $S_{0}(p)$ through $y,$ given by $c=0$ and $b=0$, namely the closed embeddings of formal schemes

[TABLE]

The map $\Omega_{S/k}|_{x}\to\Omega_{\mathfrak{W}/k}|_{y}$ maps $ds\mapsto db,\,\,\,dr\mapsto 0.$ The map $\Omega_{S/k}|_{x}\to\Omega_{\mathfrak{Z}/k}|_{y}$ is identically 0.

Proof.

The map $\mathcal{\widehat{O}}_{\widetilde{\mathscr{S}},\widetilde{x}}\to\mathcal{\widehat{O}}_{\mathscr{S}_{0}(p),y}$ is an isomorphism even before we reduce these rings modulo $p.$ Indeed, both are $3$-dimensional complete regular local rings, and the map between them induces an isomorphism on the cotangent spaces $\mathfrak{m}/\mathfrak{m}^{2}$, hence is an isomorphism. Here we use the fact that the map between cotangent spaces coincides with the corresponding map on the local models, which happens to be an isomorphism.

The two branches of $\text{Spf}(\mathcal{\widehat{O}}_{S_{0}(p),y})$ can be read off the reduction modulo $p$ of the local model $\mathbf{L}_{y}.$ As both branches are smooth over $k$, and so is the base $S$ at $x,$ the maps on cotangent spaces are easily calculated from the local models. ∎

3.3.3. Second example

For our second example assume that $x$ is an ssp point and $y$ is such that $H\simeq\kappa\otimes\alpha_{p}$ and $\zeta^{p+1}=-1$ (case I.1.a in [Bel]). In this case $(\alpha,\beta,\gamma)=(2,2,2)$ and we may assume that

[TABLE]

The most general deformation satisfying (a) (b) and (c) of Theorem 3.4 is given by

[TABLE]

[TABLE]

where $r,s,a,b,c\in\mathfrak{m}_{R}$ satisfy the relations

[TABLE]

The local models are therefore

[TABLE]

and the maps between them are given by $c\mapsto ra$, $s\mapsto-rb.$ Modulo $p$th powers of the maximal ideals these are also the maps between the completed local rings of the Picard modular surfaces at the corresponding points.

4. The global structure of $S_{0}(p)$

As before, fix an algebraic closure $k$ of $\kappa.$ In this section we concentrate on the structure of the geometric special fiber $S_{0}(p)$ over $k.$

4.1. The $\mu$-ordinary strata

4.1.1. Lots of Frobenii

Let $Y=S_{0}(p),$ and let

[TABLE]

be its base change under the Frobenius of $k.$ This is a fine moduli space for tuples $(\underline{A}_{1},H_{1})$ as in the moduli problem $(S_{0}(p))$ except that the signature of the $\mathcal{O}_{E}$-action on the Lie algebra of $A_{1}$ is now $(1,2)$ rather than $(2,1).$

This $Y^{\sigma}$ carries the universal abelian variety $\mathcal{A}_{1}=\mathcal{A}^{\sigma}=\Phi_{k}^{*}\mathcal{A}.$ It should be distinguished from $\mathcal{A}^{(p)}=\Phi_{Y}^{*}\mathcal{A},$ which lies over $Y$. The same remark and notation applies to the universal subgroup scheme $H.$ The following diagram illustrates the situation.

[TABLE]

The three squares are Cartesian. The composition of the arrows in the three top rows are the maps $\Phi_{\mathcal{A}}$, $\Phi_{Y}$ and $\Phi_{k}.$

Consider now an $R$-valued point $\xi:Spec(R)\to Y$ and let $A=\xi^{*}\mathcal{A}$ be the abelian scheme over $Spec(R)$ represented by $\xi$ (we suppress the role of $H$ and the PEL structure). Consider

[TABLE]

Then

[TABLE]

In the moduli-problem language this means that for $(\underline{A},H)\in Y(R)$

[TABLE]

The Frobenius $Fr_{A/R}$ is an isogeny $Fr_{A/R}:A\to A^{(p)}$. All of the above holds (forgetting the group $H$) also for $S$ instead of $S_{0}(p).$

4.1.2. The $\mu$-ordinary strata

We study the part of $S_{0}(p)$ lying over $S_{\mu}$, together with the map $\pi.$ Recall that we work over the algebraically closed field $k$. We are motivated by the familiar diagram of maps of modular curves (which takes advantage of the fact that $X_{0}(p)$ is defined over $\mathbb{F}_{p})$

[TABLE]

where $\pi(A,H)=A,\,\,\,\,\overline{\pi}(A_{1},H_{1})=A_{1}/H_{1}$ and $\rho(A)=(A^{(p)},A^{(p)}[{\rm Ver}]$).

Theorem 4.1**.**

(i)* Let $Y_{\mu}=\pi^{-1}(S_{\mu})\subset S_{0}(p).$ Then $Y_{\mu}$ is the disjoint union of two open sets $Y_{m}$ and $Y_{et}.$ A point $(\underline{A},H)\in S_{0}(p)(k)$ lies on $Y_{m}$ if and only if $H\simeq\kappa\otimes\mu_{p}$, and on $Y_{et}$ if and only if $H\simeq\kappa\otimes\mathbb{Z}/p\mathbb{Z}$.*

*(ii) *The map $\pi\colon Y_{\mu}\to S_{\mu}$ is finite flat of degree $p^{3}+1$. Restricted to $Y_{m}$ it yields an isomorphism

[TABLE]

Its inverse is the section

[TABLE]

cf. the proof below for the notation.

*(iii) *Consider next $Y_{et}$ and its base change $Y_{et}^{\sigma}$ under the Frobenius of $k$. Let (\underline{A}_{1},H_{1})\in$$Y_{et}^{\sigma}(R) for some $k$-algebra $R$. Then there exists a point $\underline{A}\in S_{\mu}(R)$ such that $\underline{A}_{1}\simeq\underline{A}^{(p)}=\Phi_{R}^{*}\underline{A}$. In fact, let

[TABLE]

where $H_{1}^{\perp}$ is the annihilator of $H_{1}$ under the pairing $e_{p\phi_{1}}$ on $A_{1}[p]$. Then $K_{1}$ is a finite flat, maximal isotropic, $\mathcal{O}_{E}$-stable subgroup scheme of $A_{1}[p]$. Let $B=A_{1}/K_{1},$ and descend the polarization, endomorphisms, and level-$N$ structure from $A_{1}$ to $B.$ Then

[TABLE]

so we may take $\underline{A}=\left\langle p\right\rangle^{-1}\underline{B}$. Moreover, under the isomorphism $\underline{A}_{1}\simeq\underline{A}^{(p)}$

[TABLE]

*(iv) *Restricted to $Y_{et},$ $\pi$ yields a map $\pi_{et}$, which is of degree $p^{3}$ and totally ramified, i.e. $1-1$ on $k$-points. It factors as

[TABLE]

where $Fr_{Y/k}\colon Y_{et}\to Y_{et}^{\sigma}$ is the relative Frobenius morphism, and $\overline{\pi}_{et}\colon Y_{et}^{\sigma}\to S_{\mu}$ is totally ramified of degree $p.$

In fact, identify $Y_{et}^{\sigma}$ with the moduli space for tuples $(\underline{A}_{1},H_{1})$ as before. Let $K_{1}$ and $\underline{A}$ be as in part (iii). Then the following holds:

[TABLE]

In addition, if $(\underline{A}_{1},H_{1})=Fr_{Y/k}((\underline{A},H))=(\underline{A}^{(p)},H^{(p)})$ for some $(\underline{A},H)\in Y_{\mu}(R)$, then $K_{1}=A^{(p)}[{\rm Ver}].$

*(v) *For any $R$-valued point $\underline{A}$ of $S_{\mu},$ $H={\rm Fr}(A^{(p)}[{\rm Ver}])$ is a finite flat, rank $p^{2},$ isotropic, Raynaud subgroup scheme of $A^{(p^{2})}[p]$. Furthermore, it is étale. Define a map

[TABLE]

by setting

[TABLE]

Then $\rho_{et}$ is finite flat and totally ramified of degree $p$. We have

[TABLE]

The following diagram summarizes what was said about the maps $\pi_{et}$,$\overline{\pi}_{et},\rho_{et}$.

[TABLE]

Proof.

(i) Let $Y_{\mu}=\pi^{-1}(S_{\mu}).$ This is an open subset of $S_{0}(p).$ If $R$ is any $k$-algebra and $\underline{A}\in S_{\mu}(R),$ then the group scheme $A[p]_{/R}$ admits a canonical filtration by finite flat $\mathcal{O}_{E}$-subgroup schemes

[TABLE]

Here $Fil^{1}$ is the maximal connected subgroup-scheme and is of rank $p^{4},$ while $Fil^{2}$ is the maximal subgroup scheme of multiplicative type (connected, with étale Cartier dual), and is of rank $p^{2}.$ It is also equal to the annihilator of $Fil^{1}$ under the pairing $e_{p\phi}.$ Moreover, the graded pieces are rigid in formal neighborhoods. This means that over any Artinian neighborhood $Spec(R)$ of a point, we have isomorphisms ($gr^{i}=Fil^{i}/Fil^{i+1})$

[TABLE]

as $R$-group schemes with $\mathcal{O}_{E}$-action. We remark that the filtration and the rigidity of its graded pieces hold for the whole $p$-divisible group. If $R=k$ (or any other perfect field), $A[p]$ splits canonically as the product of the three graded pieces. As these are pairwise non-isomorphic, the only rank-$p^{2}$ $\mathcal{O}_{E}$-subgroup schemes of $A[p]$ are then the unique copies of $\kappa\otimes\mu_{p},$ $\mathfrak{G}[p]_{\Sigma}$ or $\kappa\otimes\mathbb{Z}/p\mathbb{Z}$ in it. They are all Raynaud. Only the first and the last are isotropic for the Weil pairing. Thus, if $x\in S_{\mu}(k),$ there are only two points of $Y_{\mu}(k)$ above $x.$ We call $Y_{m}$ the component of $Y_{\mu}$ containing the $k$-points $(\underline{A},H)$ where $H\simeq\kappa\otimes\mu_{p},$ and $Y_{et}$ the component containing the $k$-points where $H\simeq\kappa\otimes\mathbb{Z}/p\mathbb{Z}.$ That these are indeed connected components follows from the above mentioned rigidity.

(ii) Let

[TABLE]

be the morphism defined on $R$-points $(R$ any $k$-algebra) by $\underline{A}\mapsto(\underline{A},A[p]^{m}).$ It is a section of the map $\pi,$ both $\pi\circ\sigma_{m}$ and $\sigma_{m}\circ\pi$ are the identity maps, hence $\pi$ induces an isomorphism on $Y_{m}$.

This is not the case on $Y_{et},$ as we can not split the filtration of $A[p]$ functorially over arbitrary $k$-algebra, only over perfect fields. Let us prove that $\pi_{et}:Y_{et}\to S_{\mu}$ is finite flat and totally ramified of degree $p^{3}.$ It follows from the computations of the completed local rings in §3.2 that $Y_{et}$ is non-singular. The map $\pi_{et}$ is quasi-finite and proper (see Proposition 1.5), hence finite. Any finite surjective morphism between non-singular varieties is automatically flat ([Eis] 18.17). In fact, the same argument, using regularity of the arithmetic schemes, proves that on the scheme $\mathscr{S}_{0}(p)^{\prime}$ obtained by removing $Y_{ss}=\pi^{-1}(S_{ss})$ from the special fiber of $\mathscr{S}_{0}(p)$, the map $\pi$ is finite flat to $\mathscr{S}^{\prime}=\mathscr{S}-S_{ss}.$ Since the degree in the generic fiber is $p^{3}+1$, so must be the degree in the special fiber. Since $\pi$ was shown to be an isomorphism on $Y_{m}$, on $Y_{et}$ it is finite flat of degree $p^{3},$ and of course, totally ramified ($1-1$ on geometric points). For another proof see [Bel] III.3.5.12.

(iii,iv) Since $Y_{et}^{\sigma}$ is reduced, every $R$-point of $Y_{et}^{\sigma}$ is a base-change of an $R^{\prime}$-point under a homomorphism $R^{\prime}\to R,$ where $R^{\prime}$ is reduced. We may therefore assume in the proof of (iii) and (iv) that $R$ is reduced.

We begin by showing that if $(\underline{A}_{1},H_{1})$ is an $R$-point of $Y_{et}^{\sigma},$ then $K_{1}=H_{1}+H_{1}^{\perp}[{\rm Fr}]$ is a finite flat subgroup-scheme of rank $p^{3}$ contained in $A_{1}[p].$ It is enough to prove this for the universal abelian scheme $\mathcal{A}_{1}$ over $Y_{et}^{\sigma}$, and its universal subgroup $H_{1}.$ We use the criterion for flatness, saying that if $f:X^{\prime}\to X$ is a finite morphism of schemes, $X$ is reduced, and all the fibers of $f$ have the same rank, then $f$ is also flat ([Mu], p.432). By the open-ness of the flat locus of a morphism, if $X$ is a variety over a field $k,$ it is enough to check the constancy of the fiber rank at closed points of $X$. We shall use this criterion here for group schemes over $Y_{et}^{\sigma}$, noting that the base is a non-singular variety. First, $H_{1}^{\perp}$ is clearly finite flat of rank $p^{4}$ over $Y_{et}^{\sigma}$ and $H_{1}^{\perp}[{\rm Fr}]=H_{1}^{\perp}\cap\mathcal{A}_{1}[{\rm Fr}]$ is a closed, hence finite, subgroup scheme. Its fiber rank (over the closed points of $Y_{et}^{\sigma}$!) is constantly $p$, so it is also flat. Next, $H_{1}\cap H_{1}^{\perp}[{\rm Fr}]=H_{1}[{\rm Fr}]=0$. Thus, as a subgroup functor of $\mathcal{A}_{1}[p],$

[TABLE]

is a finite flat group scheme of rank $p^{3}$.

Define $\overline{\pi}_{et}$ to be the morphism sending $(\underline{A}_{1},H_{1})\in Y_{et}^{\sigma}(R)$ to $\left\langle p\right\rangle^{-1}\underline{B}$ where $B=A_{1}/K_{1}.$ The type of ${\rm Lie}(B)$ will now be $(2,1)$, as can be easily checked. Since $K_{1}$ is a maximal isotropic subgroup scheme for the Weil pairing on $A_{1}[p],$ the polarization $p\phi_{1}$ on $A_{1}$ descends to a principal polarization of $B.$ The tame level-$N$ structure on $A_{1}$ gives rise to a tame level-$N$ structure on $B$. This completes the definition of $\overline{\pi}_{et}.$

If $(\underline{A}_{1},H_{1})=(\underline{A}^{(p)},H^{(p)})$ for $(\underline{A},H)\in Y_{et}(R)$, and $R$ is reduced, then $K_{1}$ is of rank $p^{3}$ and killed by ${\rm Ver}$, as can be checked fiber-by-fiber. This shows that

[TABLE]

hence $A_{1}/K_{1}\simeq A$ via ${\rm Ver}:A^{(p)}\to A.$ The polarization $p\phi_{1}$ descends back to $\phi$ because $\phi_{1}=\phi^{(p)}.$ Finally, if

[TABLE]

is the level-$N$ structure on $A$ and $\eta_{1}=\eta^{(p)},$ then

[TABLE]

concluding the proof that $\overline{\pi}_{et}(\underline{A}_{1},H_{1})=\underline{A}.$ This holds in particular when $R=k,$ which is enough to prove

[TABLE]

We remark that for a reduced $R$, to conclude that $K_{1}=A^{(p)}[{\rm Ver}]$ we did not have to know that $H_{1}$ was of the form $H^{(p)},$ only that $A_{1}=A^{(p)}.$ Caution must be exercised when $R$ is non-reduced though, because it is then possible to have $A^{(p)}\simeq B^{(p)}$ without $A\simeq B$. The isogeny ${\rm Ver}$ should be labeled by $A$ or $B$, and the given isomorphism between $A^{(p)}$ and $B^{(p)}$ may not carry $\ker({\rm Ver}_{A})$ to $\ker({\rm Ver}_{B})$.

In general, applying the same argument to $(A_{1}^{(p)},H_{1}^{(p)})$ implies that

[TABLE]

so

[TABLE]

By the remark above, $K_{1}=B^{(p)}[{\rm Ver}].$ We emphasize, however, that the group $H_{1}$ need not be a Frobenius base change of a similar subgroup of $B.$ To guarantee that the level-$N$ structures also match we have to twist $\underline{B}$ by the diamond operator $\left\langle p\right\rangle^{-1}$ and set $\underline{A}=\left\langle p\right\rangle^{-1}\underline{B}.$ Then $\underline{A}_{1}\simeq\underline{A}^{(p)}.$

(v) The finite subgroup scheme $\mathcal{A}^{(p)}[{\rm Fr}]\cap\mathcal{A}^{(p)}[{\rm Ver}]$ is flat over $S_{\mu}$, as it has constant fiber rank $p$ and the base is reduced. The image

[TABLE]

is isomorphic to the quotient of $\mathcal{A}^{(p)}[{\rm Ver}]$ by $\mathcal{A}^{(p)}[{\rm Fr}]\cap\mathcal{A}^{(p)}[{\rm Ver}],$ hence is also finite and flat of rank $p^{2}.$ It is isotropic, $\mathcal{O}_{E}$-stable and Raynaud. By base change from the universal case, for any $R$-valued point $\underline{A}$ of $S_{\mu},$ $H={\rm Fr}(A^{(p)}[{\rm Ver}])$ is a finite flat, rank $p^{2},$ isotropic, Raynaud subgroup scheme of $A^{(p^{2})}[p]$. It is easily seen to be étale. Since $\rho_{et}$ is defined functorially in terms of the moduli problem, it is a well-defined morphism.

It is enough to verify the equality $\rho_{et}\circ\pi_{et}=Fr_{Y/k}^{2}$ on $k$-valued points $(\underline{A},H)\in Y_{et}(k),$ namely that

[TABLE]

but if $A$ is $\mu$-ordinary this is clear. The relation $\rho_{et}\circ\overline{\pi}_{et}=Fr_{Y^{\sigma}/k}$ follows from $\rho_{et}\circ\pi_{et}=Fr_{Y/k}^{2}$ since $\pi_{et}=\overline{\pi}_{et}\circ Fr_{Y/k}$ and $Fr_{Y/k}$ is faithfully flat. The remaining assertions on $\rho_{et}$ also follow from this relation. ∎

Corollary 4.2**.**

Over $Y_{et}$ the universal abelian scheme $\mathcal{A}\simeq\mathcal{A}_{1}^{(p)}=Y_{et}\times_{\Phi_{Y},Y_{et}}\mathcal{A}_{1}$ for another abelian scheme $\mathcal{A}_{1}$ of type $(1,2).$

Proof.

In part (iii) of the theorem we showed the same for the universal abelian variety $\mathcal{A}_{1}$ over $Y_{et}^{\sigma}.$ The corollary follows by base-changing back to $Y_{et}$, or by repeating the arguments throughout with type $(1,2)$ replacing type $(2,1).$ ∎

4.1.3. A lemma on ramification

Before we continue our study of $Y_{\mu}$ we need the following result.

Lemma 4.3**.**

Let $\pi:Y\to X$ be a finite flat totally ramified morphism of degree $p$ between non-singular surfaces over $k$, an algebraically closed field of characteristic $p$. Let $\pi(y)=x.$ Then there exist local parameters $u,v$ at $y\in Y$ so that $\pi^{*}:\mathcal{\widehat{O}}_{X,x}\hookrightarrow\mathcal{\widehat{O}}_{Y,y}$ is

[TABLE]

The class of $u^{p}$ modulo $\mathfrak{\widehat{m}}_{X,x}^{2}$ spans $\ker(\pi^{*}:\Omega_{X/k}|_{x}\to\Omega_{Y/k}|_{y}),$ and is therefore independent of any choice.

Proof.

See [Ru-Sh] Theorem 4, and the Corollary at the bottom of p. 1215 there. ∎

Definition**.**

We call the line in $T_{x}X$ which is the annihilator of $\ker(\pi^{*}:\Omega_{X/k}|_{x}\to\Omega_{Y/k}|_{y})$ the unramified direction at $x$, and denote it by $T_{x}X^{ur}.$* *Then $TX^{ur}$ is a line sub-bundle of $TX.$

If $C\subset X$ is a non-singular curve such that for every $x\in C$

[TABLE]

(an integral curve for $TX^{ur}$), then $\pi:\pi^{-1}(C)^{red}\to C$ is indeed unramified, hence an isomorphism, because $\pi^{*}$ is injective on

[TABLE]

4.1.4. The unramified direction of $\overline{\pi}_{et}$

The morphism $\pi_{et}$ is “too ramified”, and we study it via the factorization $\pi_{et}=\overline{\pi}_{et}\circ Fr_{Y/k}.$ Since $\overline{\pi}_{et}$ is of degree $p,$ it admits, as we have just seen, an “unramified direction”. In §2.2 we have defined the special sub-bundle $TS^{+}$ in $TS$ outside the superspecial locus. We shall now show that over $S_{\mu}$ it coincides with the sub-bundle of unramified directions for $\overline{\pi}_{et}$. Thus the latter can be defined intrinsically in terms of the automorphic vector bundles on $S,$ without any reference to the covering $\pi.$

Theorem 4.4**.**

Let $x=\pi_{et}(y)=\overline{\pi}_{et}(y^{(p)})\in S_{\mu}.$ The unramified direction at $x$ for the map $\overline{\pi}_{et}$ is $T_{x}S^{+}.$ Equivalently, under the Kodaira-Spencer isomorphism

[TABLE]

Proof.

More precisely, we need to prove that over $Y_{et}^{\sigma}$

[TABLE]

In parts (iii) and (iv) of Theorem 4.1 we have seen that if we denote by $\mathcal{A}_{1}$ the universal abelian scheme over $Y_{et}^{\sigma}$ then $\mathcal{A}_{1}=\mathcal{B}^{(p)}$, where $\mathcal{B}=\overline{\pi}_{et}^{*}\mathcal{A},$ and the morphism $\overline{\pi}_{et}$ is induced from ${\rm Ver}:\mathcal{A}_{1}\to\mathcal{B}$, followed by $\left\langle p\right\rangle^{-1}$ on the level-$N$ structure.

Consider the abelian scheme $\mathcal{C}=\mathcal{A}_{1}/\mathcal{H}_{1}$ (over $Y_{et}^{\sigma}$) where $\mathcal{H}_{1}$ is the universal étale subgroup scheme of $\mathcal{A}_{1}.$ The isogeny** ${\rm Ver}:\mathcal{A}_{1}\to\mathcal{B}$** factors as

[TABLE]

where $\psi$ is the isogeny with kernel $\mathcal{H}_{1}$ and $\varphi$ the isogeny with kernel $\mathcal{A}_{1}[{\rm Ver}]/\mathcal{H}_{1}.$ Notice that although ${\rm Ver}:\mathcal{A}_{1}\to\mathcal{B}$ is pulled back from a similar isogeny over $S_{\mu},$ only over $Y_{et}^{\sigma}$ does it factor through $\mathcal{C}$ because $\mathcal{H}_{1}$ is *not *the pull-back of a group scheme on $S_{\mu}.$ Consider now the diagram

[TABLE]

resulting from the functoriality of the Kodaira-Spencer maps with regard to the isogeny $\varphi.$ Here ${\rm KS}_{\mathcal{B}}$ is the Kodaira-Spencer map for the family $\mathcal{B}\to Y_{et}^{\sigma}$ and likewise for $\mathcal{C}$. Note that as $\mathcal{B}=\overline{\pi}_{et}^{*}\mathcal{A}$, ${\rm KS}_{\mathcal{B}}$ is the composition of the isomorphism

[TABLE]

(we identify $\mathcal{L=\omega_{\mathcal{A}}}(\overline{\Sigma})$ with $\omega_{\mathcal{A}^{t}}(\Sigma)$ via the polarization as usual) and the map induced by

[TABLE]

The kernel of the left vertical arrow $\varphi^{*}$ is precisely $\overline{\pi}_{et}^{*}(\mathcal{P}_{0}).$ On the right hand side, however, $1\otimes(\varphi^{t})^{*\vee}$ is injective. This stems from the fact that the type of $\mathcal{C}$ (an étale quotient of $\mathcal{A}_{1}$) is $(1,2)$ while the type of $\mathcal{B}$ is $(2,1)$. Thus the type of $\mathcal{C}^{t}$ is $(2,1)$ and that of $\mathcal{B}^{t}$ (1,2). The map $(\varphi^{t})^{*}$ being surjective on the $\Sigma$-part of the cotangent spaces, its dual is injective.

We conclude that ${\rm KS}_{\mathcal{B}}(\overline{\pi}_{et}^{*}(\mathcal{P}_{0}))=0,$ hence

[TABLE]

As both sides are line bundles which are direct summands of the locally free rank 2 sheaf $\overline{\pi}_{et}^{*}\Omega_{S_{\mu}},$ the inclusion is an equality between line sub-bundles, as desired. Their annihilators in $TS$ are the “special sub-bundle” $TS^{+}$ and the “line-bundle of unramified directions” $TS^{ur}$, hence these two are also equal. ∎

In the next section we shall see that the theorem extends to the gss locus. In fact, the same proof applies, once we extend the morphism $\overline{\pi}_{et}$ and the factorization $\pi_{et}=\overline{\pi}_{et}\circ Fr_{Y/k}.$ See the proof of Theorem 4.5 (iii).

4.2. The gss strata

Recall that the supersingular locus $S_{ss}\subset S$ is the union of Fermat curves crossing transversally at the superspecial locus $S_{ssp}.$ The complement of these crossing points was denoted $S_{gss}$ and is therefore a disjoint union of open Fermat curves. In this section we study its pre-image under the morphism $S_{0}(p)\to S$ and show that it is a $\mathbb{P}^{1}$-bundle, intersecting transversally with the horizontal components of $S_{0}(p).$ Understanding the pre-image of $S_{ssp}$ will be taken up in the next section.

4.2.1. The $\mathbb{P}^{1}$-bundles

Theorem 4.5**.**

*(i) *Let $Y_{gss}=\pi^{-1}(S_{gss})^{red}.$ Then $Y_{gss}$ has the structure of a $\mathbb{P}^{1}$-bundle over the non-singular curve $S_{gss},$ with two distinguished non-intersecting non-singular curves

[TABLE]

A point $y=(\underline{A},H)\in S_{0}(p)(k)$ lies on $Z_{et}$ if and only if $H\simeq\alpha_{p^{2},\Sigma}$ and on $Z_{m}$ if and only if $H\simeq\alpha_{p^{2},\Sigma}^{*}.$ The fiber $\pi^{-1}(x)$ $(x\in S_{gss}(k))$ intersects each of the curves $Z_{et}$ or $Z_{m}$ at a unique point. At all other $k$-points $(\underline{A},H)$ of $Y_{gss},$ the group $H\simeq\mathfrak{G}[p]_{\Sigma}$.

*(ii) *The closure $\overline{Y}_{m}$ of $Y_{m}$ intersects $Y_{gss}$ transversally in $Z_{m}$. Let $Y_{m}^{\dagger}=Y_{m}\cup Z_{m}$, a locally closed subscheme of $S_{0}(p)$, and $S_{\mu}^{\dagger}=S_{\mu}\cup S_{gss}.$ Then $Y_{m}^{\dagger}$ is a non-singular surface. The map $\pi_{m}\colon Y_{m}^{\dagger}\overset{\sim}{\to}S_{\mu}^{\dagger}$ is an isomorphism, and the section $\sigma_{m}:S_{\mu}\to Y_{m}$ extends to a section of $\pi_{m}$ over $S_{\mu}^{\dagger}.$

(iii)* The closure $\overline{Y}_{et}$ of $Y_{et}$ intersects $Y_{gss}$ transversally in $Z_{et}$. Let $Y_{et}^{\dagger}=Y_{et}\cup Z_{et},$ a locally closed subscheme of $S_{0}(p).$ Then $Y_{et}^{\dagger}$ is a non-singular surface. The morphism $\overline{\pi}_{et}$ of Theorem 4.1 extends to a morphism*

[TABLE]

which is finite flat totally ramified of degree $p.$ The factorization $\pi_{et}=\overline{\pi}_{et}\circ Fr_{Y/k}$ extends to $Y_{et}^{\dagger}.$

Restricted to $Z_{et}$ the map $\pi_{et}$ is totally ramified of degree $p$ and $\overline{\pi}_{Z}=\overline{\pi}_{et}|_{Z_{et}^{\sigma}}$ is an isomorphism from $Z_{et}^{\sigma}$ onto $S_{gss}.$

*(iv) Setting ***

[TABLE]

extends the map $\rho_{et}$ to a finite flat totally ramified map of degree $p$ from $S_{\mu}^{\dagger}$ to $Y_{et}^{\dagger}.$ We have

[TABLE]

The proof of the theorem will be given in the next subsection. We caution the reader that the scheme-theoretic pre-image of $S_{gss}$ under $\overline{\pi}_{et}$ is not reduced. It is rather a nilpotent thickening of degree $p$ of the reduced curve $Z_{et}^{\sigma}$ in $Y_{et}^{\dagger\sigma}.$ Similarly the scheme-theoretic pre-image $\pi^{-1}(S_{gss})$ is non-reduced along $Z_{et}$, and only there.

We also caution that the formula (4.1) giving $\overline{\pi}_{et}$ on $Y_{et}^{\sigma}$ is no longer valid for its continuous extension to $Z_{et}^{\sigma}$. The group functor $H_{1}+H_{1}^{\perp}[{\rm Fr}]$ is represented by a finite flat group scheme on each of $Y_{et}^{\sigma}$ and $Z_{et}^{\sigma}$ separately, but even though the ranks of these group schemes are the same ($p^{3})$, they do not glue to give a group scheme over the whole of $Y_{et}^{\dagger\sigma}$. Indeed, at a closed point of $Y_{et}^{\sigma}$ this group is the kernel of ${\rm Ver}$, but this does not hold at closed points of $Z_{et}^{\sigma}.$666If $H_{1}$ and $H_{2}$ are finite flat subgroup schemes of a finite flat group scheme $G,$ then $H_{1}\cap H_{2}$ is a finite subgroup scheme, but is not necessarily flat. If it is flat, then the sum $H_{1}+H_{2},$ being isomorphic as a group functor to $H_{1}\times H_{2}/(H_{1}\cap H_{2}),$ is again represented by a finite flat group scheme. In general, however, the group-functor-quotient of a finite flat group scheme by a closed (hence finite) non-flat subgroup scheme, need not be represented by a group scheme at all, let alone by a finite flat group scheme. Thus the sum of two subgroup schemes need not be a group scheme!

The following diagram summarizes what the extensions of the maps $\pi_{et},\overline{\pi}_{et},\rho_{et}$ to the gss strata look like.

[TABLE]

Corollary 4.6**.**

(i) The maps $\overline{\pi}_{et}$ and $\sigma_{m}$ induce an isomorphism

[TABLE]

(ii) Setting $\theta=\rho_{et}\circ\pi_{m}:Y_{m}^{\dagger}\to Y_{et}^{\dagger}$ gives a commutative diagram of totally ramified finite flat morphisms between surfaces, and similarly between embedded curves (the diagonal arrows are embeddings):

[TABLE]

The map $\theta$ is of degree $p,$ and so is $\theta|_{Z_{m}}.$ In particular, the latter factors through the Frobenius of the curve $Z_{m}$ and yields an isomorphism $Z_{m}^{\sigma}\overset{\sim}{\to}Z_{et}.$

If $Z^{\prime}_{m}$ and $Z^{\prime}_{et}$ are two $\kappa$-components of $Z_{m}$ and $Z_{et}$ (i.e. defined and irreducible over $\kappa$) which map to the same $\kappa$-component $S^{\prime}_{gss}$ of $S_{gss}$ then $\theta(Z_{m}^{\prime})=Z_{et}^{\prime}$.

Proof.

The commutativity is easily checked in terms of the moduli problem. The degrees are calculated from the fact that $\pi_{m}$ is an isomorphism, $\pi_{et}$ has degree $p^{3}$ on $Y_{et}^{\dagger}$ and degree $p$ on $Z_{et}$, while $Fr_{S/K}^{2}$ has degree $p^{4}$ on $S_{\mu}^{\dagger}$ and degree $p^{2}$ on $S_{gss}$. To summarize, in the front square we have $p^{3}\times p=p^{4}\times 1,$ and in the back square we have $p\times p=p^{2}\times 1.$ The assertion about $\kappa$-components follows from the fact that $Fr_{S/k}^{2}$ preserves these components. ∎

Remark*.*

We believe that if $N=1$ (working with stacks) the geometrically irreducible components of $S_{gss}$ are already defined over $\kappa$, hence $\theta$ exchanges the irreducible components of $Z_{m}$ and $Z_{et}$ within the same irreducible component of $Y_{gss}$. This is clearly not the case when $N>1.$ Compare with supersingular points on the modular curve $X(N).$

4.2.2. Proof of Theorem 4.5

We first quote [Bu-We], Proposition 3.6. In the notation used there, the Dieudonné module of $A[p],$ for $A$ supersingular but not superspecial, is the “Dieudonné space” $\overline{\underline{B}}(3).$ Our Dieudonné module $M$ differs from the one appearing in [Bu-We], (3.2)(2) by a “Frobenius twist”. This is because we use covariant Dieudonné theory, while [Bu-We] employs Cartier theory. See [C-C-O], Appendix B.3.10, where the first (used here) is denoted $M_{*},$ and the second (used in [Bu-We]) is denoted $E_{*}.$

Proposition 4.7**.**

Let $\underline{A}\in S_{gss}(k),$ and let $M=M(A[p])$ be the covariant Dieudonné module of $A[p].$ Then $M$ has a basis over $k$ denoted $\{e_{1},e_{2},e_{3},f_{1},f_{2},f_{3}\}$ such that

(i) $\mathcal{O}_{E}$ acts on the $e_{i}$ via $\Sigma$ and on the $f_{i}$ via $\overline{\Sigma}$.

(ii) The antisymmetric pairing induced by the principal polarization $\phi$ is given by $\left\langle e_{i},f_{j}\right\rangle=(-1)^{j}\delta_{ij},\,\,\left\langle e_{i},e_{j}\right\rangle=\left\langle f_{i},f_{j}\right\rangle=0.$

(iii) $F$ and $V$ are given by the following table:

[TABLE]

By this we mean that $Fe_{1}^{(p)}=-f_{3},\,\,Ve_{3}=f_{1}^{(p)}$, etc. In particular, ${\rm Lie}(A)=M[V]=\left\langle e_{1},e_{2},f_{3}\right\rangle$.

Let $\underline{A}\in S_{\mu}^{\dagger}(R),$ where $R$ is an arbitrary $k$-algebra.

Lemma 4.8**.**

The $R$-subgroup scheme $\alpha_{p}(A^{(p)})=A^{(p)}[{\rm Fr}]\cap A^{(p)}[{\rm Ver}]$ is finite flat of rank $p$, and $\mathcal{O}_{E}$-stable.

Proof.

We have already encountered the lemma when $A$ was $\mu$-ordinary. The extension to the gss stratum works the same. It is enough to prove the lemma for the universal abelian scheme $\mathcal{A}$ over $S_{\mu}^{\dagger}.$ In this case $\alpha_{p}(\mathcal{A}^{(p)})$ is clearly finite and $\mathcal{O}_{E}$-stable, and its fibers all have the same rank $p$, as follows from Proposition 4.7. Let us make this point clear, because the proposition only deals with fibers over closed points. Let $\xi$ be any point of $S_{\mu}^{\dagger}$ (not necessarily closed), and $\overline{\{\xi\}}$ its closure (a point, a curve, or an irreducible surface). By the open-ness of the flat locus there is a non-empty connected open subset $\xi\in U\subset\overline{\{\xi\}}$ such that $\alpha_{p}(\mathcal{A}^{(p)})|_{U}$ is finite and flat over $U,$ hence all its fibers, at all the geometric points of $U,$ have the same rank. But $U(k)$ is Zariski dense in $U,$ and at a $k$-point the proposition tells us that the rank is $p.$ Hence the rank is $p$ at $\xi$ as well. Since $S_{\mu}^{\dagger}$ is reduced, by [Mu], Corollary on p. 432, $\alpha_{p}(\mathcal{A}^{(p)})$ is also flat. ∎

Proposition 4.9**.**

The finite flat group scheme $A[p]_{/R}$ has a canonical filtration

[TABLE]

by finite flat group schemes, which agrees with the canonical filtration over $S_{\mu}.$ The graded pieces are $\mathcal{O}_{E}$-stable, rank $p^{2}$ and Raynaud. Furthermore, $Fil^{1}A[p]=Fil^{2}A[p]^{\perp}$ (with respect to the Weil pairing). Over $S_{gss}$ every geometric fiber of $gr^{2}A[p]$ is of type $\alpha_{p^{2},\Sigma}^{*},$ $gr^{1}A[p]$ is of type $\kappa\otimes\alpha_{p},$ and $gr^{0}A[p]$ is of type $\alpha_{p^{2},\Sigma}.$ Let $R=k$ and assume that $\underline{A}\in S_{gss}(k).$ Then, with the notation of Proposition 4.7,

[TABLE]

We remark that unlike $\mu$-ordinary abelian varieties, the above filtration does not split, even if $R=k.$ As we shall see, $A[p]$ does not admit a subgroup scheme of type $\kappa\otimes\alpha_{p}$ at all, and while it does admit a unique subgroup scheme of type $\alpha_{p^{2},\Sigma}$, this subgroup scheme is contained in $Fil^{1}A[p],$ so does not lift $gr^{0}A[p].$

Proof.

Define

[TABLE]

This image exists because it is a quotient by a finite flat subgroup scheme. It is a closed subgroup scheme of $A[p].$ Since $A^{(p)}[{\rm Fr}]$ is finite flat of rank $p^{3},$ the Lemma implies that $Fil^{2}A[p]$ is finite flat of rank $p^{2}.$ It is furthermore isotropic for the Weil pairing $e_{p\phi}$ on $A[p]$ associated with the principal polarization $\phi$. By Cartier duality

[TABLE]

is finite flat of rank $p^{4}.$ These group schemes are clearly $\mathcal{O}_{E}$-stable.

The remaining assertions concern the geometric fibers of $A[p],$ so we assume that $R=k.$ Over the $\mu$-ordinary locus this is the same filtration that we encountered before. Assume that we are over $S_{gss},$ and use Proposition 4.7. Let $M=M(A[p]).$ Since $F$ is induced by ${\rm Ver}$ and $V$ is induced by ${\rm Fr},$ we have to compute $F(M^{(p)}[V]).$ This turns out to be $\left\langle e_{2},f_{3}\right\rangle.$ A simple check of the table in §3.1 reveals that \text{gr^{2}=}Fil^{2}A[p] is of type $\alpha_{p^{2},\Sigma}^{*}.$ Similar computations apply to $gr^{1}$ and $gr^{0}$. ∎

We can now complete the proof of Part (i) of Theorem 4.5. From the analysis of the local models it follows that $Y_{gss}$ is a non-singular surface, mapping under the map $\pi$ to the non-singular curve $S_{gss}.$ This is clear at points where $H\simeq\mathfrak{G}[p].$ At a point $y\in Y_{gss}$ where $H\simeq\alpha_{p^{2},\Sigma}$ or $H\simeq\alpha_{p^{2},\Sigma}^{*}$ the formal neighborhood of $y$ in $S_{0}(p)$ has two non-singular analytic branches which intersect transversally. Since there are at least two irreducible components of $S_{0}(p)$ passing through $y,$ the vertical component $Y_{gss}$ and (at least) one horizontal component, we conclude that there are precisely two such components, and that they are non-singular at $y.$ In particular, $Y_{gss}$ is non-singular at $y$ too.

By the Noether-Enriques Theorem ([Bea] Theorem III.4 and Proposition III.7) it is enough to prove that for any $x\in S_{gss}(k),$ the scheme-theoretic fiber

[TABLE]

of the map $\pi:Y_{gss}\to S_{gss}$ is isomorphic to $\mathbb{P}^{1}.$ We rely on the computation of local models at points $y\in Y_{x}$ in [Bel] III.4.3.8. These show that for any $y\in Y_{x}$ the map

[TABLE]

is injective, and $\pi:Y_{gss}\to S_{gss}$ is smooth at $y.$ We do not reproduce these computations here, but remark that the most problematic points turn out to be the $y$ that lie on $Z_{et}$ (where $H\simeq\alpha_{p^{2},\Sigma}$). At such points the claim follows from §3.3.2, as the analytic branch of $S_{0}(p)$ at $y$ determined by $Y_{gss}$ is the one denoted there $\mathfrak{W},$ while $S_{gss}\subset S$ is given infinitesimally by the equation $r=0.$ $Y_{x}$ is therefore a reduced non-singular curve.

Let $M$ be the covariant Dieudonné module of $A[p],$ where $A=\mathcal{A}_{x},$ see Proposition 4.7. The fiber $Y_{x}$ represents the relative moduli problem, sending a $k$-algebra $R$ to the set of finite flat rank $p^{2}$ isotropic Raynaud $\mathcal{O}_{E}$-subgroup schemes $H\subset A_{R}[p].$ Note that since $A_{R}$ is a constant abelian scheme over $Spec(R)$ both ${\rm Fr}$ and ${\rm Ver}$ are defined on it, base-changing from $k$ to $R$ the corresponding isogenies of $A.$ Let

[TABLE]

This is a constant (finite flat) subgroup scheme of rank $p,$ and if $R=k$, its Dieudonné submodule is $\left\langle e_{2}\right\rangle.$ Let

[TABLE]

another constant (finite flat) subgroup scheme, of rank $p^{3}$. If $R=k,$ its Dieudonné submodule is $\left\langle e_{2},f_{1},f_{3}\right\rangle.$ We claim that

[TABLE]

hence classifying $H_{/R}$ is the same as classifying finite flat rank $p$ subgroups of $\beta_{p}(A_{R})/\alpha_{p}(A_{R}).$ Since $Y_{x}$ is a reduced non-singular curve, it is enough to check these inclusions when $R$ is reduced and of finite type over $k.$ Since the closed points of $Spec(R)$ are then dense, we may assume $R=k.$ But over $k$, ${\rm Ver}$ and ${\rm Fr}$ are nilpotent on $H,$ which is of rank $p^{2},$ so both ${\rm Ver}^{2}$ and ${\rm Fr}^{2}$ must kill it. On the other hand, $H$ must contain an $\alpha_{p}$-subgroup, because it is local with a local Cartier dual.

Now $\beta_{p}(A_{R})/\alpha_{p}(A_{R})$ is nothing but $\alpha_{p}^{2}$ (of type $(\overline{\Sigma},\overline{\Sigma})$) and it is well-known that the moduli problem of classifying its rank-$p$ subgroups is represented by $\mathbb{P}_{/k}^{1}.$ One checks that the isotropy and Raynaud conditions are automatically satisfied for such an $H.$

Let $R=k.$ The subgroup scheme $H$ is completely determined by its Dieudonné submodule

[TABLE]

where $\lambda=(\lambda_{1}:\lambda_{3})\in\mathbb{P}^{1}(k).$ Here $N_{0}=N_{(0:1)}=M(H)$ if $H=Fil^{2}(A[p])\simeq\alpha_{p^{2}}^{*}.$ Similarly, $N_{\infty}=N_{(1:0)}=M(H)$ where $H\simeq\alpha_{p^{2}}$ because $N_{(1:0)}$ is killed by $F$ and $V^{2}$ but not by $V.$ For all other values of $\lambda\neq 0,\infty,$ $N_{\lambda}=M(H)$ where $H$ is of type $\mathfrak{G}[p]_{\Sigma},$ because $N_{\lambda}$ is killed by $F^{2}$ and $V^{2}$ but the kernels of $F$ or $V$ are only $1$-dimensional.

Part (ii): Let us show that the totality of points $(\underline{A},H)\in Y_{gss}(k)$ where $H\simeq\alpha_{p^{2}}^{*}$, makes up a *curve $Z_{m},$ *that $\pi$ induces an isomorphism of this curve onto $S_{gss}$, and that the closure of $Y_{m}$ intersects $Y_{gss}$ transversally in this curve. For this purpose, consider the section

[TABLE]

mapping an $R$-valued point $\underline{A}$ to $(\underline{A},H)$, where $H=Fil^{2}A[p]={\rm Ver}(A^{(p)}[{\rm Fr}]).$ The image of the section is a surface isomorphic to the base, intersecting $Y_{\mu}$ in its connected component $Y_{m}$ and $Y_{gss}$ in the curve $Z_{m}.$ Finally, the transversality of the intersection of the closure of $Y_{m}$ and $Y_{gss}$ follows from the calculation of the completed local ring of $S_{0}(p)$ at a point $y\in Z_{m},$ see §3.2.

Part (iii): We turn our attention to the points $(\underline{A},H)\in Y_{gss}(k)$ where $H\simeq\alpha_{p^{2}}.$ The condition ${\rm Ver}(H^{(p)})=0$ is a closed condition on the moduli problem $S_{0}(p)$. It is satisfied throughout $Y_{et}$ and on $Y_{gss}$ it holds precisely at the given points where $H\simeq\alpha_{p^{2}}.$ We claim that this set forms a curve $Z_{et}$, which is the intersection of the closure of $Y_{et}$ and $Y_{gss}.$ Indeed, $\pi$ being proper, the closure of $Y_{et}$ must meet every fiber $Y_{x}$ for $x\in S_{gss}(k)$, and such a fiber has a unique point where $H\simeq\alpha_{p^{2}}.$ That the intersection is transversal follows as before from §3.2.

Write $Y_{et}^{\dagger}=Y_{et}\cup Z_{et}.$ The computations in §3.2 show that $Y_{et}^{\dagger}$ is non-singular. So is $Y_{et}^{\dagger\sigma}$.

We claim that since $\pi:Y_{et}^{\dagger}\to S$ factors through $Fr_{Y/k}:Y_{et}^{\dagger}\to Y_{et}^{\dagger\sigma}$ over the dense open set $Y_{et}$, it factors through $Fr_{Y/k}$ everywhere. Indeed, consider the local ring $\mathcal{O}_{S,x}$ at $x=\pi(y)\in S$, where $y\in Y_{et}^{\dagger}$ is a closed point. Let $y^{(p)}=Fr_{Y/k}(y)\in Y_{et}^{\dagger\sigma}.$ For the function fields we have

[TABLE]

Thus $\mathcal{O}_{S,x}\subset k(Y_{et}^{\dagger})^{p}\cap\mathcal{O}_{Y_{et}^{\dagger},y}.$ But the ring on the right is just $\mathcal{O}_{Y_{et}^{\dagger\sigma},y^{(p)}}$, because $y$ is the *unique *point above $y^{(p)}$ in $Y_{et}^{\dagger}$ and $\mathcal{O}_{Y_{et}^{\dagger\sigma},y^{(p)}}$ is normal. For every affine subset $U=Spec(R)\subset Y_{et}^{\dagger}$ the ring $R$ is the intersection of all the $\mathcal{O}_{Y_{et}^{\dagger},y}$ for closed points $y\in U$, and similarly for $Fr_{Y/k}(U)\subset Y_{et}^{\dagger\sigma}.$ This proves the claim.

Thus $\overline{\pi}_{et}$ extends to a morphism from $Y_{et}^{\dagger\sigma}$ to $S_{\mu}^{\dagger}$. It is a finite morphism, because $\pi_{et}:Y_{et}^{\dagger}\to S_{\mu}^{\dagger}$ is finite. Both source and target are non-singular surfaces, so by [Eis] 18.17 it is also flat, totally ramified of degree $p$. It therefore defines a line sub-bundle $TS^{ur}$ of unramified directions in the tangent bundle there, as in Lemma 4.3, now over all of $S_{\mu}^{\dagger}.$ Recall that the special sub-bundle $TS^{+}$ was defined on the whole of $S_{\mu}^{\dagger}$ as well. The two line sub-bundles $TS^{+}$ and $TS^{ur}$ coincide over $S_{\mu}$ (Theorem 4.4), hence also over $S_{gss}$, by continuity.

As $TS^{+}$ is tangent to $S_{gss}$ along the general supersingular stratum, we get, from the discussion following Lemma 4.3, that $\overline{\pi}_{Z}\colon Z_{et}^{\sigma}\to S_{gss}$ is unramified. As it is also totally ramified (bijective on $k$-points), it is an isomorphism.

In retrospect, we can look at the factorization $\pi_{et}=\overline{\pi}_{et}\circ Fr_{Y/k}$ also from the moduli point of view as follows. Consider the abelian scheme $\mathcal{B}=\overline{\pi}_{et}^{*}\mathcal{A}$ which is the pull-back of the universal abelian scheme over $S_{\mu}^{\dagger}$ to $Y_{et}^{\dagger\sigma}.$ Consider also the universal abelian scheme $\mathcal{A}_{1}$ over $Y_{et}^{\dagger\sigma}$. Over the dense open subset $Y_{et}^{\sigma}$ $\mathcal{A}_{1}\simeq\mathcal{B}^{(p)},$ as was shown in the proof of Theorem 4.1. It follows that this relation persists over $Z_{et}^{\sigma}$, and a-fortiori we may define $\overline{\pi}_{et}$ by sending $(\underline{A}_{1},H_{1})\in Y_{et}^{\dagger\sigma}(R)$ to $\left\langle p\right\rangle^{-1}{\rm Ver}(\underline{B}^{(p)})\in S_{\mu}^{\dagger}(R).$

Part (iv): By Lemma 4.8, and the arguments used before, ${\rm Fr}(A^{(p)}[{\rm Ver}])$ is a finite flat rank-$p^{2}$ isotropic Raynaud subgroup scheme of $A^{(p^{2})}[p],$ for any $A\in S_{\mu}^{\dagger}(R),$ for any $k$-algebra $R.$ Since $\rho_{et}$ is now defined functorially in terms of the moduli problems, it is a well defined morphism. The argument is identical to the one used for the proof of Part (v) of Theorem 4.1.

Since the equality $\rho_{et}\circ\pi_{et}=Fr_{Y/k}^{2}$ has already been established on $Y_{et}=Y_{et}(k)$, it extends by continuity to $Y_{et}^{\dagger}.$ The relation $\rho_{et}\circ\overline{\pi}_{et}=Fr_{Y^{\sigma}/k}$ follows from $\rho_{et}\circ\pi_{et}=Fr_{Y/k}^{2}$ since $\pi_{et}=\overline{\pi}_{et}\circ Fr_{Y/k}$ and $Fr_{Y/k}$ is faithfully flat. The remaining assertions on $\rho_{et}$ also follow from this relation. This concludes the proof of Theorem 4.5.

4.2.3. A closer look at Example 3.3.2

It is instructive to look again at the diagram

[TABLE]

at a point $y\in Z_{et}(k).$ We have found the local models $\mathcal{\widehat{O}}_{S,x}\simeq k[[r,s]]$ and $\mathcal{\widehat{O}}_{S_{0}(p),y}\simeq k[[a,b,c]]/(bc)$. The map between the local models is

[TABLE]

This is far from the correct map between the completed local rings, which should be injective. Let $\mathcal{\widehat{O}}_{\mathfrak{W}}$ and $\mathcal{\widehat{O}}_{\mathfrak{Z}}$ be the quotients of $\mathcal{\widehat{O}}_{\mathfrak{Y}}=\mathcal{\widehat{O}}_{S_{0}(p),y}$ which were introduced in §3.3.2. The first is obtained by modding out $(c)$, and is the analytic branch determined by the inclusion $Y_{gss}\subset S_{0}(p)$. The second is obtained by modding out $(b)$, and is the analytic branch determined by the inclusion $Y_{et}^{\dagger}\subset S_{0}(p)$.

Claim 4.10*.*

The diagram $\mathcal{\widehat{O}}_{S,x}\to\mathcal{\widehat{O}}_{\mathfrak{W}}$ is isomorphic to the diagram

[TABLE]

and the diagram $\mathcal{\widehat{O}}_{S,x}\to\mathcal{\widehat{O}}_{\mathfrak{Z}}$ is isomorphic to the diagram

[TABLE]

This is more than could be deduced from the local models alone.

Proof.

After a change of variable we may assume that $r=0$ is the equation of $S_{gss}$ in a formal neighborhood of $x$ on $S.$ Therefore $r$ maps to [math] in $\mathcal{\widehat{O}}_{\mathfrak{W}}$. The local parameter $s$ projects (modulo $(r))$ to a local parameter of the curve $S_{gss}.$ We already know that it should map to $b$ modulo $p$th powers. Since $b=c=0$ is the formal equation of the curve $Z_{et}$ (the intersection of the two analytic branches) on $\mathfrak{W},$ and since the map $Z_{et}\to S_{gss}$ is purely inseparable of degree $p,$ we see that we may choose $a$ so that $s\mod(b)=a^{p}.$ A last change of variables allows us to assume that actually $s=b+a^{p}$.

The second diagram is treated similarly. Here the key point is to recall that the map $\pi_{et}$ from $\mathfrak{Z}$ to $Spf(\widehat{\mathcal{O}}_{S,x})$ factors through ${\rm Fr}$. The resulting map $\overline{\pi}_{et}$ on $\mathfrak{Z}^{\sigma}$ was shown to be of degree $p$ and unramified in the direction of $S_{gss}$. ∎

Both diagrams are compatible with $\mathcal{\widehat{O}}_{S,x}\to\mathcal{\widehat{O}}_{\mathfrak{Y}}=\mathcal{\widehat{O}}_{S_{0}(p),y}$ being given by

[TABLE]

4.3. The ssp strata

4.3.1. The superspecial combs

We now turn our attention to the superspecial strata of $S_{0}(p).$ Let $x\in S_{ssp}(k)$ and $Y_{x}=\pi^{-1}(x).$ We shall contend ourselves with the determination of the *reduced *scheme $Y_{x}^{red},$ of finite type over $k.$ The scheme theoretic pre-image of $x$ will not be reduced along the component denoted below $F_{x}$, see the discussion following the theorem.

Theorem 4.11**.**

(i) $Y_{x}^{red}$ is the union of $p+2$ projective lines, arranged as follows. One irreducible component, which we call $F_{x}$, intersects the remaining $p+1$ projective lines transversally, each at a different point $\zeta\in F_{x}.$ With a natural choice of a coordinate on $F_{x},$ this $\zeta$ can be taken to be777This is a non-trivial statement, as it has consequences for the cross ratio of the intersection points, which is independent of the chosen coordinate on the basis of the comb. a $p+1$ root of $-1.$ These $p+1$ projective lines, which we label as $G_{x}[\zeta]$, are disjoint from each other.

A point $(\underline{A},H)\in Y_{x}(k)$ lies on $F_{x}$ if and only if $H\simeq\kappa\otimes\alpha_{p}$. If this is the case, the invariant $\gamma(\underline{A},H)=1$ if $(\underline{A},H)$ lies on a non-singular point of $Y_{x}$, and is equal to $2$ if it lies at the intersection of $F_{x}$ and some $G_{x}[\zeta]$ (i.e. if it is the point $\zeta$). Finally, if $(\underline{A},H)$ lies on $G_{x}[\zeta]$ but not on $F_{x},$ the group $H\simeq\mathfrak{G}[p]_{\Sigma}.$

(ii)* The closure $\overline{Y}_{\mu}$ of $Y_{\mu}=Y_{m}\cup Y_{et}$ in $S_{0}(p)$ intersects $Y_{x}^{red}$ in $F_{x}.$*

*(iii) *Let $W$ be the closure of an irreducible component of $Y_{gss}.$ Then $W$ is a $\mathbb{P}^{1}$-bundle over an irreducible component $\mathscr{C}=\pi(W)$ of $S_{ss}$. If $x\in S_{ssp}$ and $W_{x}=W\cap Y{}_{x}^{red}$ then $W_{x}$ is one of the $G_{x}[\zeta].$ Precisely one such $W$ passes through $G_{x}[\zeta]$ for a given $x$ and $\zeta$. Thus the closures of the irreducible components of $Y_{gss}$ do not intersect each other.

(iv)* The closures of the curves $W\cap Z_{et}$ and $W\cap Z_{m}$ intersect $G_{x}[\zeta]$ at the point $\zeta=G_{x}[\zeta]\cap F_{x}.$*

See Figures 4.1, 4.2. We refer to the irreducible components $W$ of the closure of $Y_{gss}$ as the supersingular (ss) screens. We refer to the $Y_{x}$ for $x$ superspecial as the *superspecial (ssp) combs. *The component $F_{x},$ which we draw horizontally, is called the base of the comb, and the vertical components $G_{x}[\zeta]$ are called its teeth. The points $\zeta$ are called the roots of the teeth.

Proof.

(i) Let $A=\mathcal{A}_{x}.$ We first analyze what happens on the level of Dieudonné modules. Fix a model of $\mathfrak{G}_{\Sigma}$ over $k,$ let $\mathfrak{G}_{\overline{\Sigma}}=\mathfrak{G}_{\Sigma}^{\sigma}$ and fix the polarization

[TABLE]

so that the resulting pairing on $\mathfrak{G}[p]_{\Sigma}$, $(x,y)\mapsto\left\langle x,\lambda(y)\right\rangle$ is alternating. The group scheme $A[p]$ is isomorphic to

[TABLE]

so that the polarization induced on it by $\phi_{x}$ is the product $\lambda^{2}\times\lambda^{\sigma}$ of the polarizations of the three factors. Consequently [Bu-We], the polarized Dieudonné module \text{M=}M(A[p]) is given by $M=\left\langle e_{1},e_{2},e_{3},f_{1},f_{2},f_{3}\right\rangle_{k}$, where the endomorphisms act on the $e_{i}$ via $\Sigma$ and on the $f_{i}$ via $\overline{\Sigma},$ where $\left\langle e_{i},f_{j}\right\rangle=\delta_{ij},$ $\left\langle e_{i},e_{j}\right\rangle=\left\langle f_{i},f_{j}\right\rangle=0$ and where the action of $F$ and $V$ is given by the table

[TABLE]

By this we mean $Fe_{3}^{(p)}=-f_{3},\,\,\,Ve_{3}=f_{3}^{(p)}$ etc.

Let $H\subset A[p]$ be as in $(S_{0}(p))$. Since $M(H)$ is balanced we may write

[TABLE]

The conditions that have to be satisfied are $V(M(H))\subset M(H)^{(p)},\,\,\,F(M(H)^{(p)})\subset M(H),$ and the isotropy condition

[TABLE]

Observe that $M(H)$ contains $\beta_{1}^{p}e_{1}+\beta_{2}^{p}e_{2}.$ If $\alpha_{3}\neq 0$ this forces $\beta_{1}=\beta_{2}=0,$ and then the isotropy condition gives also $\beta_{3}=0,$ an absurd. Therefore $\alpha_{3}=0.$ We distinguish two cases.

Case I (the base of the comb): $\beta_{1}=\beta_{2}=0.$ This case is characterized by the fact that $M(H)$ is killed by both $V$ and $F,$ so that $H\simeq\kappa\otimes\alpha_{p}.$ We may take $\beta_{3}=1$ and $H$ is classified by

[TABLE]

Consider in this case the group $H^{\perp}/H.$ Its Dieudonné module is given by

[TABLE]

An easy check shows that $H^{\perp}/H$ is of type $\mathfrak{G}[p]_{\Sigma}$, unless $\zeta^{p+1}=-1,$ where it is of type $\kappa\otimes\alpha_{p}.$ The invariant $\gamma(\underline{A},H)=\dim_{k}{\rm Lie}(H^{\perp}/H)$ is thus 1 in the former case, and $2$ in the latter.

Case II (the teeth of the comb): $(\beta_{1}:\beta_{2})\in\mathbb{P}^{1}(k).$ Then, $\zeta=(\alpha_{1}:\alpha_{2})=(\beta_{1}^{p}:\beta_{2}^{p})$ and the isotropy condition forces

[TABLE]

i.e. $\zeta^{p+1}=-1.$ Fix $\zeta,$ hence the point $(\beta_{1}:\beta_{2}).$ The $H$ in question are classified by $\beta_{3}\in\mathbb{A}^{1}(k).$ Their $M(H)$ is killed by $V^{2}$ and $F^{2}$ but neither by $V$ nor by $F,$ so $H$ must be isomorphic to $\mathfrak{G}[p]_{\Sigma}.$ We observe that when $\beta_{3}=\infty,$ i.e. $(\beta_{1}:\beta_{2}:\beta_{3})=(0:0:1)$ we are back in Case I. This is the root of the tooth.

This analysis strongly suggests the picture outlined in Part (i), but does not quite *prove *it. To give a rigorous proof we proceed as follows. The fiber $Y_{x}$ represents the relative moduli problem assigning to any $k$-algebra $R$ the set of subgroup schemes $H\subset A_{R}[p]$ of type $(S_{0}(p))$. Observe that since $A_{R}$ is constant, both ${\rm Fr}$ and ${\rm Ver}$ are defined on it, by base change from $A.$ We let $\alpha_{p}(A_{R})=A_{R}[{\rm Fr}]\cap A_{R}[{\rm Ver}]$ and

[TABLE]

Case I. Consider first the closed locus $F_{x}\subset Y{}_{x}^{red}$ defined by

[TABLE]

Over $F_{x}$ we have $\alpha_{p}(H)=H.$ Indeed, since $F_{x}$ is a reduced curve it is enough to check the inclusion $H\subset\alpha_{p}(A_{R})$ at geometric points, where it follows from the analysis of their Dieudonné modules as above. However, $\alpha_{p}(A_{R})=\alpha_{p,\Sigma}^{2}\times\alpha_{p,\overline{\Sigma}},$ so the problem becomes that of classifying $\mathcal{O}_{E}$-subgroup schemes of type $\kappa\otimes\alpha_{p}=\alpha_{p,\Sigma}\times\alpha_{p,\overline{\Sigma}}$ in it. As the factor of type $\alpha_{p,\overline{\Sigma}}$ is unique, this is the same as classifying subgroup schemes of rank $p$ in $\alpha_{p,\Sigma}^{2},$ a problem that is represented by $\mathbb{P}_{/k}^{1}.$ This gives us the base of the comb, whose $k$-points are described in terms of their Dieudonné submodules as before.

Case II. Let $G_{x}$ be the open curve which is the complement of $F_{x}$ in $Y_{x}^{red}.$ Over $G_{x},$ the group $\alpha_{p}(H)$ is of rank $p.$ Observe that $H\cap\mathfrak{G}[p]_{\Sigma}^{2}$ is non-zero, because otherwise, via projection to the third factor, $H$ would be of type $\mathfrak{G}[p]_{\overline{\Sigma}},$ which is forbidden. It follows that $\alpha_{p}(H\cap\mathfrak{G}[p]_{\Sigma}^{2})$ is also non-zero, so must coincide with $\alpha_{p}(H).$ The $\alpha_{p}\subset\mathfrak{G}[p]_{\Sigma}^{2}$ were classified before by $\mathbb{P}_{/k}^{1}.$ Our $\alpha_{p}(H)$ is therefore classified by $\zeta=(\alpha_{1}:\alpha_{2})\in\mathbb{P}^{1}(R).$ The Dieudonné module computation above shows that $\zeta$ restricts, at every geometric point, to a $p+1$ root of $-1$. However, the equation $X^{p+1}+1=0$ is separable, so if $R$ is a local ring in characteristic $p$ and $\zeta\in R$ satisfies this equation modulo $\mathfrak{m}_{R}$, it satisfies it in $R$. This means that $\alpha_{p}(H)$ is locally constant over $Spec(R)$. There remains the classification of $H/\alpha_{p}(H),$ which sits in general “diagonally” in $(\mathfrak{G}[p]_{\Sigma}^{2}/\alpha_{p}(H))\times\mathfrak{G}[p]_{\overline{\Sigma}}.$ The same argument that was used to show that $\alpha_{p}(H)$ is constant, shows now that the projection $K$ of $H/\alpha_{p}(H)$ to $(\mathfrak{G}[p]_{\Sigma}^{2}/\alpha_{p}(H))$ is constant, and in fact is given by the point $(\beta_{1}:\beta_{2})=(\alpha_{1}^{p}:\alpha_{2}^{p})\in\mathbb{P}^{1}(R).$ The classification of $H/\alpha_{p}(H)$ is therefore the same as the classification of all the $R$-morphisms of this fixed $K$ to $\alpha_{p}(\mathfrak{G}[p]_{\overline{\Sigma}}).$ This moduli problem, of classifying morphisms from a fixed copy of $\alpha_{p}$ to another, is represented by $\mathbb{A}_{/k}^{1}.$This gives the tooth of the comb labeled $G_{x}[\zeta]$.

The two cases (I) and (II) cover $Y_{x}^{red}.$ It remains to remark that the intersection of the closure of $G_{x}[\zeta]$ with $F_{x}$ is transversal. This follows, as usual, from §3.2.

(ii) The condition ${\rm Fr}(H)=0$ is a closed condition and holds throughout $Y_{m}.$ It therefore holds also in the intersection of its closure $\overline{Y}_{m}$ with $Y_{x}.$ As this condition is not satisfied on the teeth of the comb (outside their roots), the closure $\overline{Y}_{m}$ intersects $Y_{x}$ in $F_{x}.$ The same argument, applied to the condition ${\rm Ver}(H^{(p)})=0$ proves that the closure $\overline{Y}_{et}$ of $Y_{et}$ also intersects $Y_{x}$ in $F_{x}.$ As we have previously shown that $Y_{m}^{\dagger}$ and $Y_{et}^{\dagger}$ are disjoint, $\overline{Y}_{m}$ and $\overline{Y}_{et}$ intersect only in the superspecial locus, and their intersection is the union of the $F_{x}$ for $x\in S_{ssp}.$ This intersection is transversal, as follows from the description of the completed local rings in §3.2.

(iii) The classification of the completed local rings of $S_{0}(p)$ shows that through a point $\zeta\in F_{x}$ which is not a root of a tooth (i.e. $\zeta^{p+1}\neq-1)$ pass only $2$ analytic branches. As $\overline{Y}_{et}$ and $\overline{Y}_{m}$ already account for these two analytic branches, the closure $W$ of a connected component of $Y_{gss}$ can only meet $Y_{x}$ in one of the lines $G_{x}[\zeta]$. Since the points of $G_{x}[\zeta]$ are generically non-singular on $S_{0}(p)$, exactly one such $W$ passes through every $G_{x}[\zeta]$. These $W$ are non-singular surfaces projecting to a component $\mathscr{C}$ of $S_{ss}$ and the fiber above each geometric point (including now the superspecial points) is $\mathbb{P}^{1}.$ By the Noether-Enriques theorem quoted before, they are $\mathbb{P}^{1}$-bundles.

(iv) The condition ${\rm Fr}(H)=0$ is a closed condition and holds throughout $Z_{m}.$ It therefore holds also on its closure. It follows that this closure intersects a tooth $G_{x}[\zeta]$ at its root, because points other than the root support an $H$ of type $\mathfrak{G}[p]_{\Sigma}$ which is not killed by ${\rm Fr}.$ A similar argument invoking the condition ${\rm Ver}(H^{(p)})=0$ proves that the closure of $Z_{et}$ also meets the teeth of the combs in their roots. The two curves $Z_{et}$ and $Z_{m}$, which are disjoint over the gss locus, intersect over every superspecial point.

This concludes the proof of the theorem. ∎

4.3.2. The maps to $S^{\#}$

Recall the construction of the blow-up $S^{\#}$ of $S$ at the ssp points, given in §2.3. The exceptional divisor $E_{x}$ at $x=[\underline{A}]\in S_{ssp}(k)$ classifies lines in $\mathcal{P}=\omega_{A/k}(\Sigma).$

The isomorphism $\pi_{m}:Y_{m}^{\dagger}\simeq S_{\mu}^{\dagger}$ extends to an isomorphism

[TABLE]

In terms of the moduli problems, it sends $(\underline{A},H)\in\overline{Y}_{m}(R)$ to $(\underline{A},\ker(\omega_{A/R}(\Sigma)\to\omega_{H/R}(\Sigma)).$ If $R=k,$ $A$ is $\mu$-ordinary and $H=A[p]^{m}$ then

[TABLE]

is uniquely determined by $A$. The same holds if $A$ is gss and $H=Fil^{2}A[p].$ On the other hand if $x=[\underline{A}]$ is ssp then $\mathcal{P}[V]$ is the whole of $\mathcal{P}$ and $H$ “selects” a line in it. This establishes an isomorphism

[TABLE]

From the universal property of blow-ups, the projection $\pi_{et}:\overline{Y}_{et}\to S$ also factors through a map

[TABLE]

mapping $F_{x}$ to $E_{x}.$ This map is now proper and quasi-finite, hence finite. The two surfaces are non-singular, so the map is also flat. Its degree is $p^{3}.$ We have seen that on the open dense $Y_{et}^{\dagger}$ it factors through $Fr_{Y/k},$ i.e.

[TABLE]

and this forces the map $\pi_{et}^{\#}$ to factor in the same way $\pi_{et}^{\#}=\overline{\pi}_{et}^{\#}\circ Fr_{Y/k}$ over the whole of $\overline{Y}_{et}.$ The map $\overline{\pi}_{et}^{\#}$ is finite flat totally ramified of degree $p,$ and it can be shown that it is ramified of degree $p$ along the lines $F_{x}^{\sigma}.$ Thus $\pi_{et}^{\#}$ is ramified of degree $p^{2}$ along $F_{x}$ (and of an extra degree $p$ in a normal direction).

We emphasize that $\pi_{m}^{\#}$ and $\pi_{et}^{\#}$ do not agree on $F_{x}.$ Instead, the following diagram extends the one from Corollary 4.6.

[TABLE]

The degrees of the maps in the front square (on surfaces) are $p^{3}\times p=p^{4}\times 1.$ In the back square (on projective lines) they are $p^{2}\times 1=p^{2}\times 1.$

4.3.3. How embedded modular curves meet

$F_{x}$

Let $X$ be the special fiber of the modular curve $\mathscr{X}$ which was constructed on $\mathscr{S}$ in §1.4. Consider the modular curve $\mathscr{X}_{0}(p)$ parametrizing, in addition to the triple $\underline{B}=(B,\nu,M)$, also a finite flat subgroup scheme $H_{B}\subset B[p]$ of rank $p.$ Enhance the map $\mathscr{Z}_{0}\times_{\underline{\text{Isom}}(\mathbb{Z}/N\mathbb{Z},\mu_{N})}\mathscr{X}\to\mathscr{S}$ to a map

[TABLE]

by setting $H$ to be the image of $\mathcal{O}_{E}\otimes H_{B}$ in $A(\underline{B}_{0},\underline{B})$. Note that since $H_{B}$ is automatically isotropic, and the polarization on $A$ is induced from the polarizations of $B$ and $B_{0},$ this $H$ is isotropic. It is also clearly Raynaud.

Proposition 4.12**.**

Let $X_{0}(p)$ be the special fiber of $\mathscr{X}_{0}(p).$ Let $x\in S_{ssp}(k).$ Then under the above morphism $X_{0}(p)$ meets the component $F_{x}\subset Y_{x}$ in a point $\zeta$ satisfying

[TABLE]

Thus both the supersingular screens on $S_{0}(p)$ and the modular curves cross the superspecial strata $F_{x}$ at $\mathbb{F}_{p^{2}}$-rational points, but while the supersingular screens cross at a $\zeta$ satisfying $\zeta^{p+1}=-1,$ the modular curves cross at the remaining ones.

Proof.

As we shall see in the next chapter, the $\kappa$-rational $\zeta\in F_{x}$ are characterized by the fact that $A^{\prime}=A/H$ is superspecial. At other points of $F_{x}$ this $A^{\prime}$ is supersingular of $a$-number 2, but not superspecial. For the pair $(A,H)$ that is constructed from the “elliptic curve data” on $X_{0}(p),$ it is easily seen that $A^{\prime}$ is either $\mu$-ordinary or superspecial, depending on whether $B$ is ordinary or supersingular.

Among these $\kappa$-rational points the points with $\zeta^{p+1}=-1$ are characterized by $\gamma(\underline{A},H)=2,$ i.e. the group $H^{\perp}/H=\ker(\psi)$ being isomorphic to $\kappa\otimes\alpha_{p}.$ All the rest have $\gamma=1.$ In our case, $H=\mathcal{O}_{E}\otimes H_{B}$ is maximal isotropic in $A_{1}(\underline{B})[p],$ so its annihilator in $A[p]=A_{1}[p]\times B_{0}[p]$ is $H\times B_{0}[p].$ it follows that

[TABLE]

and $\gamma=1.$ ∎

5. The structure of $\widetilde{S}$

5.1. The global structure of $\widetilde{S}$

The moduli space $\widetilde{\mathscr{S}}$ was defined in Section 1.2.3. Typically, moduli spaces involving parahoric level structure are “complicated”, and may involve issues such as non-reduced components, complicated singularities etc. It is interesting, and important for our further applications, that $\widetilde{\mathscr{S}}$ turns out to be quite simple. In essence, its special fiber is a collection of smooth surfaces intersecting transversally at a reduced non-singular curve.

5.1.1. Flatness of $\widetilde{\pi}$

The following proposition stands in sharp contrast to the non-flatness of $\pi.$ It is also key to understanding the geometry of the surface

[TABLE]

This surface, which is generically of degree $(p+1)(p^{3}+1)$ over the Picard modular surface $\mathscr{S}$, “is” the geometrization of the Hecke operator $T_{p}.$ We intend to study it in a future work.

Proposition 5.1**.**

The morphism $\widetilde{\pi}:\mathscr{S}_{0}(p)\to\mathscr{\widetilde{S}}$ is finite flat of degree $p+1$.

Proof.

Both arithmetic surfaces are regular. The map $\widetilde{\pi}$ is proper, and, as we shall see below, analyzing its geometric fibers one-by-one, also quasi-finite. It is therefore finite. By [Eis], 18.17, it is flat. The degree can be read off in characteristic 0. ∎

From now on we concentrate on the structure of the geometric special fiber $\widetilde{S}_{k}$ of $\mathscr{\widetilde{S}}$ over $k,$ and omit the subscript $k.$ We study $\widetilde{S}$ together with the map

[TABLE]

and make strong use of the facts that we have already established for $S_{0}(p).$

5.1.2. The fibers of $\widetilde{\pi}$

To study the geometric fibers of $\pi$ we had to study, for a given $\underline{A},$ the subgroup schemes $H\subset A[p]$ for which $(\underline{A},H)\in S_{0}(p)(k).$ This was achieved by analyzing $M(A[p])$ and its $2$-dimensional, isotropic, balanced $\mathcal{O}_{E}$-stable Dieudonné submodules. To study the geometric fibers of $\widetilde{\pi}$ we have to look, for a given $\underline{A}^{\prime},$ for all the possible $(\underline{A},H)$ yielding $\underline{A}^{\prime}$ upon the process of dividing by $H$ and descending the polarization. Equivalently, by Proposition 1.4, we have to look for all the subgroup schemes $J$ such that $(\underline{A}^{\prime},J)\in\widetilde{S}_{0}(p)(k).$ This reduces the computation of the fibers of $\widetilde{\pi}$ to Dieudonné-module computations, as was the case with $\pi.$ However, starting with one $(\underline{A},H)$ mapping under $\widetilde{\pi}$ to $\underline{A}^{\prime}$, finding all the others in the fiber above $\underline{A}^{\prime}$ requires in general the knowledge of $M(A[p^{2}])$ and not only of $M(A[p]).$ This makes the following sections technically more complicated than the previous ones.

5.1.3. The stratification of $\widetilde{S}$

We suppress $(\iota^{\prime},\eta^{\prime})$ from the notation and refer to $R$-points of $\widetilde{S}$ ($R$ a $k$-algebra) as $(A^{\prime},\psi).$ Given $(A^{\prime},\psi)\in\widetilde{S}(k)$ the subgroup scheme

[TABLE]

is of rank $p^{2},$ self-dual (i.e. isomorphic to its Cartier dual), stable under $\iota^{\prime}(\mathcal{O}_{E})$ and Raynaud. Its Lie algebra ${\rm Lie}(\ker(\psi))$ is 1 or 2-dimensional888If it were 0-dimensional, $A^{\prime}$ would be $\mu$-ordinary and $\ker(\psi)\simeq\kappa\otimes\mathbb{Z}/p\mathbb{Z},$ but this group is not self-dual., and carries an action of $\kappa.$ We call its type the type (or signature) of $\ker(\psi)$ and denote it by $\tau(\psi).$ Similarly the maximal $\alpha_{p}$-subgroup of $A^{\prime}[p]$ is of rank $p,p^{2}$ or $p^{3}$, and the $\kappa$-type of its Lie algebra is called the $a$-type of $A^{\prime}$, and denoted $a(A^{\prime}).$

Theorem 5.2**.**

(i)** **The surface $\widetilde{S}$ is the union of 7 disjoint, locally closed, nonsingular strata $\widetilde{S}_{*}[**]$, as shown in the table. The name of each stratum indicates the type of $A^{\prime}_{\widetilde{x}}$ for $\widetilde{x}$ in the stratum ($\mu$-ordinary, gss or ssp), and, in brackets, the type of $\ker(\psi)$. The last column indicates what types of $(\underline{A},H)$ lie in $\widetilde{\pi}^{-1}(\widetilde{x})$. The first entry in the last column refers to the stratum of $S$ in which $\underline{A}$ lies. The second refers to the type of $H$ ($\mathfrak{G}$ stands for $\mathfrak{G}[p]$). If $\underline{A}$ is ssp there is a third entry, which we now explain.

Recall that the ssp strata of $S_{0}(p)$ are unions of projective lines admitting a natural coordinate $\zeta$. The third entry refers to $\zeta$. Depending on whether $\zeta\in\mathbb{F}_{p^{2}}$ or not, and in the case of the components $F_{x},$ also on whether it is a $p+1$ root of $-1$, $\widetilde{\pi}(A,H)$ may land in different strata of $\widetilde{S}.$

[TABLE]

(ii)* The closure relations between the various strata are described by the following diagram, where an arrow $X\to Y$ indicates specialization, i.e. that $Y\subset\overline{X}$.*

[TABLE]

The strata $\widetilde{S}_{gss}[\Sigma,\overline{\Sigma}]$ and $\widetilde{S}_{ssp}[\Sigma,\overline{\Sigma}]$ are singular on $\widetilde{S},$ and the rest are nonsingular.

See Figure 5.1

Proof.

The invariants $(\tau(\psi),a(A^{\prime}))$ characterize the stratum in $\widetilde{S}$, and the seven cases in the last column are mutually exclusive and exhaustive. It is therefore enough to verify that starting with a point $(\underline{A},H)\in S_{0}(p)(k)$ in a prescribed stratum of $S_{0}(p)$, we end up with the right pair of invariants $(\tau(\psi),a(A^{\prime}))$. For this we use the covariant Dieudonné module $M(A[p^{\infty}]).$

(1) If $A$ is $\mu$-ordinary, so is $A^{\prime},$ and vice versa. As in this case

[TABLE]

and $H$ is either $\mathcal{O}_{E}\otimes\mu_{p}$ or $\mathcal{O}_{E}\otimes\mathbb{Z}/p\mathbb{Z}$, H^{\perp}/H\simeq\mathfrak{G}[p]_{\text{\Sigma}} so $\tau(\psi)=\Sigma$. Since upon dividing by $H$ we get $A^{\prime}[p^{\infty}]\simeq A[p^{\infty}]$, $a(A^{\prime})=\Sigma$. The map $Y_{m}\to\widetilde{S}_{\mu}$ is surjective, purely inseparable of degree $p,$ while $Y_{et}\to\widetilde{S}_{\mu}$ is an isomorphism. This follows from the following two facts: (a) $Y_{\mu}\to\widetilde{S}_{\mu}$ is finite flat of degree $p+1,$ (b) If $y\in Y_{et}(k)$ then $\widetilde{\pi}$ is étale at $y,$ while if $y\in Y_{m}(k)$ it is ramified there (see §3.2). We conclude that if $\widetilde{x}\in\widetilde{S}_{\mu}(k)$ the fiber $\widetilde{\pi}^{-1}(\widetilde{x})$ contains precisely 2 points. Alternatively, we could have used the model $\widetilde{S}_{0}(p)$ (see §1.2.6) to show that there are precisely two possibilities for $J$ to go with an $\underline{A}^{\prime}\in\widetilde{S}_{\mu}(k).$

(2) Assume next that $A$ is gss and $H\simeq\mathfrak{G}[p].$ The analysis of $H^{\perp}/H$ is easy, since $H^{\perp}\subset A[p],$ so we can use Proposition 4.7. With the notation used there

[TABLE]

for some $(\alpha_{1}:\alpha_{2})\neq 0,\infty.$ It follows that $M(H^{\perp}/H)=\left\langle\alpha_{2}\overline{e}_{1}-\alpha_{1}\overline{e}_{3},\overline{f}_{1}\right\rangle$ where the bar denotes the class modulo $M(H).$ Since this space is killed by $V^{2}$ and $F^{2}$ but neither by $F$ nor by $V,$ $H^{\perp}/H\simeq\mathfrak{G}[p].$ Since $M(H^{\perp}/H)[V]=\left\langle\overline{f}_{1}\right\rangle$, ${\rm Lie}(H^{\perp}/H)$ is of type $\overline{\Sigma}.$

To analyze the $\alpha_{p}$-subgroup of $A^{\prime}$ and conclude that it is of rank $p^{2}$ and type $(\Sigma,\overline{\Sigma})$, we need to know $M(A[p^{2}])$. This, unlike $M(A[p])$, depends on the particular $A$, and not only on it being of type gss. The computations needed to verify this are deferred to the appendix.

(3) Assume that $A$ is gss and $H\simeq\alpha_{p^{2},\Sigma}.$ Using the notation of Proposition 4.7

[TABLE]

so $M(H^{\perp}/H)=\left\langle\overline{e}_{3},\overline{f}_{3}\right\rangle.$ This module is killed by both $V$ and $F$ so $H^{\perp}/H\simeq\kappa\otimes\alpha_{p},$ and its Lie algebra is of type $(\Sigma,\overline{\Sigma}).$ The computation of $a(A^{\prime})$ is again deferred to the appendix. The case $A$ gss and $H\simeq\alpha_{p^{2},\Sigma}^{*}$ is treated similarly.

(4) Assume that $A$ is ssp. Then the covariant Dieudonné module $M=M(A[p^{\infty}])$ is freely spanned over $W(k)$ by a basis $e_{1},e_{2},e_{3},f_{1},f_{2},f_{3}$ satisfying (i) $\mathcal{O}_{E}$ acts on the $e_{i}$ via $\Sigma$ and on the $f_{i}$ via $\overline{\Sigma}$ (ii) $\left\langle e_{i},f_{j}\right\rangle=\delta_{ij}$, $\left\langle e_{i},e_{j}\right\rangle=\left\langle f_{i},f_{j}\right\rangle=0$ (iii) the action of $F$ and $V$ is given by the table

[TABLE]

.

See [Bu-We], Lemma (4.1) and [Vo], Lemma 4.2. Note that Vollaard works over $W(\kappa)$ and uses a slightly different normalization, but over $W(k)$ her model and the one above become isomorphic. Let $\overline{M}=M/pM=M(A[p])$ (called in [Bu-We] the Dieudonné space) and denote by $\overline{e}_{i}$ and $\overline{f}_{i}$ the images of the basis elements. Using the notation of the proof of Theorem 4.11, we distinguish two cases.

*Case I *(the base of the comb): In this case $H$ is of type $\kappa\otimes\alpha_{p}$ and

[TABLE]

As we have seen in the proof of Theorem 4.11, $H^{\perp}/H=\ker(\psi)$ is of type $\mathfrak{G}[p]_{\Sigma}$, unless $\zeta=(\alpha_{1}:\alpha_{2})$ satisfies $\zeta^{p+1}=-1,$ where it is of type $\kappa\otimes\alpha_{p}.$ This gives the entries for $\tau(\psi)$ in rows 4,6 and 7 of the table. We proceed to compute the $a$-number and $a$-type of $A^{\prime}$. For this observe that $M^{\prime}=M(A^{\prime}[p^{\infty}])$ sits in an exact sequence

[TABLE]

hence inside the isocrystal $M_{\mathbb{Q}}$

[TABLE]

Here we let $\widetilde{\alpha}_{i}$ denote any element of $W(k)$ mapping to $\alpha_{i}$ modulo $p$. To compute the Dieudonné module of the $\alpha_{p}$-subgroup of $A^{\prime}$ we must compute

[TABLE]

The kernel of $V$ on $M^{\prime}/pM^{\prime}$ is spanned over $k$ by the images of the vectors $\{e_{1},e_{2},e_{3},\widetilde{\alpha}_{1}^{\sigma}f_{1}+\widetilde{\alpha}_{2}^{\sigma}f_{2}\}$ where $\sigma$ is the Frobenius on $W(k).$ Similarly, the kernel of $F$ is spanned by the images of $\{e_{1},e_{2},e_{3},\widetilde{\alpha}_{1}^{\sigma^{-1}}f_{1}+\widetilde{\alpha}_{2}^{\sigma^{-1}}f_{2}\}$. The span of $\{e_{1},e_{2},e_{3}\}$ in $M^{\prime}/pM^{\prime}$ is two dimensional and of type $\Sigma,\Sigma.$ We see that if $\zeta=(\alpha_{1}:\alpha_{2})\notin\mathbb{F}_{p^{2}}$ then $\alpha_{p}(A^{\prime})$ is of rank $p^{2}$, hence $A^{\prime}$ is gss (supersingular but not superspecial), and $a(A^{\prime})=\{\Sigma,\Sigma\}.$ On the other hand if $\zeta\in\mathbb{F}_{p^{2}}$ then $\alpha_{p}(A^{\prime})$ is of rank $p^{3},$ so $A^{\prime}$ is superspecial, and $a(A^{\prime})=\{\Sigma,\Sigma,\overline{\Sigma}\}.$ This completes the verification of $\tau(\psi)$ and $a(A^{\prime})$ in rows 4,6 and 7 of the table.

*Case II *(the teeth of the comb): In this case $H$ is of type $\mathfrak{G}[p]_{\Sigma},$

[TABLE]

where $\zeta=(\alpha_{1}:\alpha_{2})=(\beta_{1}^{p}:\beta_{2}^{p})$ satisfies $\zeta^{p+1}=-1$ and $\beta_{3}\in k$ is arbitrary. Now $M(H^{\perp}/H)$ is spanned by the images of $-\beta_{3}\overline{e}_{1}+\beta_{1}\overline{e}_{3}$ and $\overline{f}_{3}$ modulo $M(H)$, so $H^{\perp}/H=\ker(\psi)$ is seen to be of type $\mathfrak{G}[p]_{\overline{\Sigma}}$. This confirms the invariant $\tau(\psi)$ in rows 2 and 5 of the table. Regarding $a(A^{\prime})$ we compute, as in Case I, $M^{\prime}=M(A^{\prime}[p^{\infty}]):$

[TABLE]

We find that $M^{\prime}/pM^{\prime}[V]$ is spanned over $k$ by the images of

[TABLE]

Note that $p^{-1}(\widetilde{\beta}_{1}^{\sigma}e_{1}+\widetilde{\beta}_{2}^{\sigma}e_{2})\in M^{\prime}$ because of the relation $(\alpha_{1}:\alpha_{2})=(\beta_{1}^{p}:\beta_{2}^{p}).$ Likewise $M^{\prime}/pM^{\prime}[F]$ is spanned over $k$ by the images of

[TABLE]

Now $\widetilde{\alpha}_{1}^{\sigma^{-1}}f_{1}+\widetilde{\alpha}_{2}^{\sigma^{-1}}f_{2}$ and $\widetilde{\alpha}_{1}^{\sigma}f_{1}+\widetilde{\alpha}_{2}^{\sigma}f_{2}$ both represent the class of $\widetilde{\beta}_{1}f_{1}+\widetilde{\beta}_{2}f_{2}$ in $M^{\prime}/pM^{\prime}$. Similarly $p^{-1}(\widetilde{\beta}_{1}^{\sigma}e_{1}+\widetilde{\beta}_{2}^{\sigma}e_{2})$ and $p^{-1}(\widetilde{\beta}_{1}^{\sigma^{-1}}e_{1}+\widetilde{\beta}_{2}^{\sigma^{-1}}e_{2})$ both represent the class of $p^{-1}(\widetilde{\alpha}_{1}e_{1}+\widetilde{\alpha}_{2}e_{2})$ in $M^{\prime}/pM^{\prime}.$ It follows that the span of $f_{3}$ and $\widetilde{\alpha}_{1}^{\sigma}f_{1}+\widetilde{\alpha}_{2}^{\sigma}f_{2}$ in $M^{\prime}/pM^{\prime}$ is 1-dimensional and of type $\overline{\Sigma}$. Regarding the $\Sigma$-component of $M^{\prime}/pM^{\prime}[V]\cap M^{\prime}/pM^{\prime}[F]$, $e_{1}$ and $e_{2}$ contribute a 1-dimensional piece there. If $\beta_{3}\in\mathbb{F}_{p^{2}}$ then $p^{-1}(\widetilde{\beta}_{1}^{\sigma}e_{1}+\widetilde{\beta}_{2}^{\sigma}e_{2})+\widetilde{\beta}_{3}^{\sigma}e_{3}$ and $p^{-1}(\widetilde{\beta}_{1}^{\sigma^{-1}}e_{1}+\widetilde{\beta}_{2}^{\sigma^{-1}}e_{2})+\widetilde{\beta}_{3}^{\sigma^{-1}}e_{3}$ contribute another 1-dimensional piece, but otherwise they do not agree modulo $pM^{\prime}.$

To sum up, if $\beta_{3}\notin\mathbb{F}_{p^{2}}$ then $A^{\prime}$ is gss and $a(A^{\prime})=\{\Sigma,\overline{\Sigma}\}.$ If $\beta_{3}\in\mathbb{F}_{p^{2}}$ then $A^{\prime}$ is ssp and $a(A^{\prime})=\{\Sigma,\Sigma,\overline{\Sigma}\}$. This completes the verification of $\tau(\psi)$ and $a(A^{\prime})$ in rows 2 and 5.

Since the morphism $\widetilde{\pi}$ is finite flat of degree $p+1,$ the dimensions of the strata of $\widetilde{S}$ follow from the known dimensions of the strata of $S_{0}(p).$ Moreover, each geometric fiber has $p+1$ points if one counts multiplicities. We have already noted that the map $Y_{m}\to\widetilde{S}_{\mu}$ is surjective, purely inseparable of degree $p,$ while $Y_{et}\to\widetilde{S}_{\mu}$ is an isomorphism. This proves that for $\widetilde{x}\in\widetilde{S}_{\mu}(k),$ $\#\widetilde{\pi}^{-1}(\widetilde{x})=2,$ but it also proves that for $\widetilde{x}\in\widetilde{S}_{gss}[\Sigma,\overline{\Sigma}](k)$ we have $\#\widetilde{\pi}^{-1}(\widetilde{x})=2$. Indeed, such a point must have pre-images both in $Z_{et}$ and in $Z_{m}$ but the morphism $\widetilde{\pi}:Y_{m}\to\widetilde{S}$ being totally ramified and 1:1 on geometric points, must extend to a totally ramified morphism on $Z_{m}$, since the ramification locus is closed. Thus $\widetilde{\pi}$ is 1:1 on $Z_{m}(k).$ It is clearly 1:1 on $Z_{et}(k)$ because it is an isomorphism on $Z_{et}$.

Similar arguments show that $\widetilde{\pi}$ is totally ramified of degree $p+1$ on the base of the comb denoted $F_{x}$ in Theorem 4.11, where $A$ is ssp and $H$ of type $\kappa\otimes\alpha_{p}$. This shows that $\#\widetilde{\pi}^{-1}(\widetilde{x})=1$ in rows 4,6 and 7 of the table.

Finally, at a generic point $y$ lying on a tooth of a comb or on the gss screens (i.e. where $A$ is ssp or gss but $H$ is of type $\mathfrak{G}[p]_{\Sigma}$) $\widetilde{\pi}$ induces an isomorphism on the completed local rings as can be seen from the table in Proposition 3.6, hence is étale. It follows that the image of such a point has $p+1$ distinct pre-images.

This concludes the proof of part (i) of the theorem. Part (ii) follows from the relations between the closures of the pre-images of the seven strata in $S_{0}(p).$ ∎

5.2. Analysis of $\widetilde{\pi}$

5.2.1. Analysis of $\widetilde{\pi}$ along the $\mu$-ordinary strata

We denote by $\widetilde{\pi}_{et}$ and $\widetilde{\pi}_{m}$ the restrictions of $\widetilde{\pi}$ to $Y_{et}$ (or even $Y_{et}^{\dagger}$) and $Y_{m}$ (or $Y_{m}^{\dagger}$).

Proposition 5.3**.**

*(i) *The map $\widetilde{\pi}_{et}\colon Y_{et}\overset{\sim}{\to}\widetilde{S}_{\mu}$ is an isomorphism. Denote by

[TABLE]

the section which is its inverse. If $\underline{A}^{\prime}\in\widetilde{S}_{\mu}(R)$ then $A^{\prime}[{\rm Fr}]+\ker(\psi)$ is a finite flat subgroup $J$ satisfying the conditions listed in Proposition 1.4, $p\psi$ descends to a principal polarization $\phi$ on $A^{\prime}/J$ and

[TABLE]

(ii) The map $\widetilde{\pi}_{m}:Y_{m}\to\widetilde{S}_{\mu}$ is finite flat totally ramified of degree $p.$**

Proof.

We have already seen that $\widetilde{\pi}_{et}$ is an isomorphism and that $\widetilde{\pi}_{m}$ is a finite flat totally ramified map of degree $p.$ It remains to check the assertion about $\widetilde{\sigma}_{et}.$ Let us first check the claims made about $J$. As usual, by reduction to the universal object, we may assume that $R$ is reduced. Then $A^{\prime}[Fr]\cap\ker(\psi)$ is a finite group scheme over $R,$ all of whose fibers have the same rank $p,$ so is finite flat, and

[TABLE]

is finite flat of rank $p^{4}.$ It is also maximal isotropic for $e_{p\psi},$ $\mathcal{O}_{E}$-stable and $J/\ker(\psi)$ is Raynaud. All these statements are checked fiber-by-fiber. We may therefore descend $p\psi$ to a principal polarization of $A^{\prime}/J$ and form the tuple $\widetilde{\sigma}_{et}(\underline{A}^{\prime}).$ It is now a simple matter to check that if $A^{\prime}=A/H$ where $(\underline{A},H)\in Y_{et}(k)$ then

[TABLE]

and $A^{\prime}[p]/J=p^{-1}H/A[p]$ gets mapped back to $H.$ When we add level-$N$ structure twisted by the diamond operator $\left\langle p\right\rangle^{-1}$ to the definition of $\widetilde{\sigma}_{et}(\underline{A}^{\prime})$ we ensure that $\widetilde{\sigma}_{et}$ is indeed the inverse of $\widetilde{\pi}_{et}.$ ∎

The next corollary follows directly from the definitions of the various maps and we omit its proof.

Corollary 5.4**.**

(i)** **On $R$-points of the moduli problems the maps

[TABLE]

are given by

[TABLE]

Their compositions are the maps $Fr^{2}\colon S_{\mu}\to S_{\mu}^{(p^{2})}=S_{\mu}$ or $Fr^{2}\colon\widetilde{S}_{\mu}\to\widetilde{S}_{\mu}^{(p^{2})}=\widetilde{S}_{\mu}$ (here we use the fact that $S$ and $\widetilde{S}$ are defined over $\kappa$).

(ii)** **The maps

[TABLE]

are given by

[TABLE]

5.2.2. Analysis of $\widetilde{\pi}$ along the curves $Z_{et}$ and $Z_{m}$

Proposition 5.5**.**

Let $\widetilde{Z}$ be the stratum $\widetilde{S}_{gss}[\Sigma,\overline{\Sigma}].$ The morphism $\widetilde{\pi}_{et}:Z_{et}\to\widetilde{Z}$ is an isomorphism. The morphism $\widetilde{\pi}_{m}:Z_{m}\to\widetilde{Z}$ is totally ramified of degree $p.$

Proof.

Let $Y_{et}^{\dagger}=Y_{et}\cup Z_{et}$ and $\widetilde{S}_{\mu}^{\dagger}=\widetilde{S}_{\mu}\cup\widetilde{Z}$. The map $\widetilde{\pi}_{et}:Y_{et}^{\dagger}\to\widetilde{S}_{\mu}^{\dagger}$ is finite, and induces an isomorphism between the open dense subsets $Y_{et}\simeq\widetilde{S}_{\mu}.$ From the classification of the completed local rings in Proposition 3.6 it follows that $\widetilde{S}_{\mu}^{\dagger}$ is smooth, hence its local rings are integrally closed and $\widetilde{\pi}_{et}$ is an isomorphism. A similar argument shows that $\widetilde{\pi}_{m}:Y_{m}^{\dagger}\to\widetilde{S}_{\mu}^{\dagger}$ is finite flat totally ramified of degree $p,$ where $Y_{m}^{\dagger}=Y_{m}\cup Z_{m}.$

In principle, the unramified direction (see Lemma 4.3) for $\widetilde{\pi}_{m}:Y_{m}^{\dagger}\to\widetilde{S}_{\mu}^{\dagger}$ at a point $\widetilde{x}\in\widetilde{Z}$ could be transversal to $\widetilde{Z}$ or tangential to it. We claim that it is everywhere transversal, i.e. the schematic pre-image of $\widetilde{Z}$ is $Z_{m}$ (with its reduced structure) but $\widetilde{\pi}|_{Z_{m}}$ is totally ramified of degree $p.$ This can be seen in a variety of ways.999Were the unramified direction everywhere tangential to $\widetilde{Z}$, the schematic pre-image of $\widetilde{Z}$ would be a nilpotent thickening of order $p$ of $Z_{m}$, but $\widetilde{\pi}$ would be an isomorphism on the reduced curve. In general, of course, there is also a “mixed option”, where the unramified direction is generically transversal, but tangential to $\widetilde{Z}$ at finitely many points. We shall deduce it from Corollary 5.4. Observe first that the maps $j_{et}$ and $j_{m}$ extend to similarly denoted maps

[TABLE]

and may then be restricted to the gss curves $\widetilde{Z}$ and $S_{gss}.$ The claim follows now from the following established facts: (a) $\sigma_{m}:S_{gss}\simeq Z_{m}$ and $\widetilde{\sigma}_{et}:\widetilde{Z}\simeq Z_{et}$ are isomorphisms, (b) $\pi_{et}:Z_{et}\to S_{gss}$ is totally ramified of degree $p$ (equivalently, $\overline{\pi}_{et}:Z_{et}^{(p)}\simeq S_{gss}$ is an isomorphism) (c) $j_{et}\circ j_{m}=Fr^{2}$ hence, restricted to the curve $S_{gss}$, it is totally ramified of degree $p^{2}.$ ∎

The same argument used to show that $\widetilde{\pi}_{et}$ extends to an isomorphism on $Y_{et}^{\dagger}$, and that $\widetilde{\pi}_{m}$ extends to a totally ramified map on $Y_{m}^{\dagger}$ gives the following.

Proposition 5.6**.**

Let $\overline{Y}_{et}$ and $\overline{Y}_{m}$ denote the closures of $Y_{et}$ and $Y_{m}$ in $S_{0}(p)$. Then $\widetilde{\pi}_{et}$ extends to an isomorphism from $\overline{Y}_{et}$ to the closure $\overline{\widetilde{S}}_{\mu}$ of $\widetilde{S}_{\mu}.$ The map $\widetilde{\pi}_{m}$ extends to a totally ramified map of degree $p$ from $\overline{Y}_{m}$ to $\overline{\widetilde{S}}_{\mu}$.

A computation similar to the above, that we leave out, yields the following.

Corollary 5.7**.**

Let $\theta:Y_{m}^{\dagger}\to Y_{et}^{\dagger}$ be the map $\theta=\rho_{et}\circ\pi_{m}$ (see Corollary 4.6). Then

[TABLE]

5.2.3. Analysis of $\widetilde{\pi}$ along the gss screens $Y_{gss}$

Let $W$ be an irreducible component of the closure $\overline{Y}_{gss}$ of $Y_{gss}.$ As we have seen in Theorem 4.11, these irreducible components are smooth $\mathbb{P}^{1}$-bundles over Fermat curves, and do not intersect each other. Outside (the closure of) $Z_{et}$ and $Z_{m}$ the restriction of $\widetilde{\pi}$ to $W,$ which we denote from now on $\widetilde{\pi}_{W},$ is étale. It is also étale at $y\in Z_{et}(k).$ This follows from §3.3.2.

(1) We have

[TABLE]

Proof: $\widetilde{\pi}(\overline{Z}_{et}\cap W)$ is an irreducible component of $\overline{\widetilde{Z}}$, the closure of the stratum $\widetilde{Z}=\widetilde{S}_{gss}[\Sigma,\overline{\Sigma}]$. So is $\widetilde{\pi}(\overline{Z}_{m}\cap W$). The two intersect at the image of any point $\zeta$ which is “a base of a tooth of a comb”, points where $\overline{Z}_{et}$ and $\overline{Z}_{m}$ meet. Since the irreducible components of $\overline{\widetilde{Z}}$ are disjoint, the two components coincide.

(2) We have

[TABLE]

Proof: this follows from (1) since $\widetilde{\pi}^{-1}(\overline{\widetilde{Z}})=\overline{Z}_{et}\cup\overline{Z}_{m}.$

(3) Let $W,W^{\prime}$ be two components of $\overline{Y}_{gss}$. Then $\widetilde{\pi}(W)\cap\widetilde{\pi}(W^{\prime})=\emptyset.$

Proof: Each $\widetilde{\pi}(W)$ is an irreducible component of $\widetilde{\pi}(\overline{Y}_{gss})$. But the irreducible components of $\widetilde{\pi}(\overline{Y}_{gss})$ are disjoint from each other and are uniquely determined by their intersection with $\overline{\widetilde{S}}_{\mu}$, i.e. with $\overline{\widetilde{Z}}$. The claim follows from (2), since $\widetilde{\pi}(\overline{Z}_{et}\cap W)\cap\widetilde{\pi}(\overline{Z}_{et}\cap W^{\prime})=\emptyset$, as $\widetilde{\pi}_{et}$ is an isomorphism.

(4) We give another proof of (5.1). It is based on the following lemma, which is of independent interest. Recall that $S$ is defined over $\kappa=\mathbb{F}_{p^{2}},$ although we consider it over $k=\mathbb{\overline{F}}_{p^{2}}.$ It follows that $Gal(k/\kappa)$ permutes the irreducible components of $S_{gss}.$ The diamond operators also act on these irreducible components.

Lemma 5.8**.**

Let $Z$ be an irreducible component of $S_{gss}.$ Then $Fr_{p^{2}}(Z)=\left\langle p\right\rangle(Z).$

Proof.

For the proof of the lemma we may increase $N.$ Indeed, if $N|N^{\prime}$ and $Z,Z^{\prime}$ are as above for $N$ and $N^{\prime},$ with $Z^{\prime}$ mapping to $Z,$ then the validity of the lemma for $Z^{\prime}$ implies it for $Z.$ Since the closure $\overline{Z}$ of every irreducible component of $S_{gss}$ contains at least two superspecial points, and since when $N$ is large enough, through any two superspecial points passes at most one such $\overline{Z}$ [Vo], it is enough to prove that for $x\in S_{ssp}(k)$

[TABLE]

Let $x=(A,\phi,\iota,\eta).$ Every supersingular elliptic curve $B$ over $k$ has a model $B_{0}$ over $\kappa$, whose Frobenius of degree $p^{2}$ satisfies

[TABLE]

By the Tate-Honda theorem [Ta], all the endomorphisms of $B$ are already defined over $\kappa$. We may therefore assume that $A\simeq B^{3}$ and $\iota$ are defined over $\kappa.$ Since $A$ admits at least one principal polarization defined over $\kappa,$ and its endomorphisms are all defined over $\kappa,$ $\phi$ is defined over $\kappa.$ Thus $\left(A,\phi,\iota\right)$ is invariant under $Fr_{p^{2}}.$ But the relation $Fr_{p^{2}}=p$ on $A[N]$ means that $Fr_{p^{2}}(\eta)=\left\langle p\right\rangle\circ\eta,$ which concludes the proof. ∎

Now use the relation

[TABLE]

from Corollary 5.4, and its extension to $S_{\mu}^{\dagger}$ from the proof of Proposition 5.5. The left hand side fixes the irreducible components of $S_{gss}$, hence also the irreducible components $W$ of $\overline{Y}_{gss}.$ Let $y\in\overline{Z}_{m}\cap W$. Then $y^{\prime}=\widetilde{\sigma}_{et}\circ\widetilde{\pi}_{m}(y)\in\overline{Z}_{et}\cap W,$ or

[TABLE]

This shows that $\widetilde{\pi}(\overline{Z}_{m}\cap W)=\widetilde{\pi}(\overline{Z}_{et}\cap W)$ as was to be shown.

(5) The map $\widetilde{\pi}_{W}:W\to\widetilde{\pi}(W)$ is finite flat of degree $p+1$.

*Proof: *This follows from (3) since $\widetilde{\pi}$ in the large is finite flat of degree $p+1$.

We next want to analyze how $\widetilde{\pi}$ is behaved when restricted to a fiber $W_{x}=\pi^{-1}(x)$ of $\pi$ above a gss point $x$. Recall that $W_{x}\simeq\mathbb{P}^{1}.$

(6) Let $y_{m}$ and $y_{et}$ be the unique points on $Z_{m}\cap W_{x}$ and $Z_{et}\cap W_{x}$ respectively. Then $\widetilde{\pi}(y_{m})\neq\widetilde{\pi}(y_{et}).$

Proof: Equivalently, we have to show that the images under $\pi$ of $y$ and $y^{\prime}$ as in (5.2), which are in the same fiber for $\widetilde{\pi},$ are distinct. But $\pi(y^{\prime})=\left\langle p\right\rangle^{-1}\pi(y)^{(p^{2})}.$ We claim that if $\pi(y)=(A,\phi,\iota,\eta)$ then already $(A,\phi,\iota)$ is not defined over $\kappa,$ so is not isomorphic to $(A^{(p^{2})},\phi^{(p^{2})},\iota^{(p^{2})}).$ This follows from the fact, established in [Vo], that when $N=1$ any irreducible curve $Z$ in the supersingular locus of the coarse moduli space associated with the algebraic stack $S$ is defined over $\kappa$, and is birationally isomorphic to the Fermat curve

[TABLE]

Let $\mathcal{\mathscr{C}}\to Z$ be the normalization of $Z$. This $\mathcal{\mathscr{C}}$ has $p^{3}+1$ $\kappa$-rational points, which are precisely the points mapping to superspecial points on $Z.$ Furthermore, all the self-intersections of $Z$ are at $\kappa$-rational points. It follows that no $x\in Z(k)$ which is gss is fixed under $Fr_{p}^{2}.$ Since the diamond operators do not affect $(A,\phi,\iota)$, *a fortiori *$\pi(y^{\prime})\neq\pi(y).$

Starting with $x=x^{(1)}\in S_{gss}(k)$ we may now form a sequence of points $x^{(1)},\dots,x^{(r)}$ such that if $y_{m}^{(i)}$ and $y_{et}^{(i)}$ are the respective points on $W_{x^{(i)}}$ then

[TABLE]

This sequence becomes periodic after $d$ steps, where $d$ is the minimal number so that $\left\langle p\right\rangle^{-d}\circ Fr_{p}^{2d}(x)=x.$

(7) The map $\widetilde{\pi}:W_{x}\to\widetilde{\pi}(W_{x})$ is a birational isomorphism.

*Proof: *We have to show that the map is generically 1-1. For that it is enough to find a single point $y\in W_{x}$ so that $\widetilde{\pi}$ is étale at $y$ and $\widetilde{\pi}^{-1}(\widetilde{\pi}(y))=\{y\}.$ In view of (6), the unique point on $Z_{et}\cap W_{x}$ is such a point.

We do not answer the question whether $\widetilde{\pi}$ is everywhere 1-1. We summarize the discussion of this section in the following theorem.

Theorem 5.9**.**

The map $\widetilde{\pi}$ induces a bijection between the vertical irreducible components of $\widetilde{S}$ and of $S_{0}(p)$. The map $\pi$ induces a bijection between the vertical irreducible components of $S_{0}(p)$ and the irreducible components of the curve $S_{ss}.$ The vertical irreducible components of $\widetilde{S}$ are mutually disjoint. Let $W$ be a vertical irreducible component of $S_{0}(p).$ Then $\widetilde{\pi}_{W}$ is finite flat of degree $p+1$ and is étale outside $W\cap\overline{Z}_{m}.$ The restriction of $\widetilde{\pi}_{W}$ to $W_{x}=\pi^{-1}(x)$ for $x\in S_{gss}$ is a birational isomorphism and maps the unique intersection points of $W_{x}$ with $Z_{et}$ and $Z_{m}$ to distinct points.

6. Appendix

6.1. The classification of the gss Dieudonné modules

In the appendix we perform some computations on the covariant Dieudonné module of a gss abelian variety. We first recall their classification, following Vollaard [Vo].

Fix $\delta\in\mu_{p^{2}-1}\subset W(\kappa)\subset W(\kappa)_{\mathbb{Q}}=E_{p}$ such that

[TABLE]

Let $\mathbf{M}$ be the free $W(\kappa)$-module on $e_{1},e_{2},e_{3},f_{1},f_{2},f_{3}$ and let $\mathcal{O}_{E}$ act on the $e_{i}$ via $\Sigma$ (the canonical embedding of $E$ in $E_{p})$ and on the $f_{i}$ via $\overline{\Sigma}.$ Let $F$ be the $\sigma$-linear endomorphism101010In the appendix we depart from our habit of writing $F$ as a linear map from $\mathbf{M}^{(p)}$ to $\mathbf{M}.$ of $\mathbf{M}$ whose matrix w.r.t. the above basis is

[TABLE]

i.e. F(e_{1})=pf_{1},$$F(e_{2})=f_{2},\dots,F(f_{3})=e_{3}. Let $V$ be the $\sigma^{-1}$-linear endomorphism with the same matrix. Note that $\tau=V^{-1}F$ is the identity on $\mathbf{M}.$ Let $\mathbf{M}_{k}=W(k)\otimes_{W(\kappa)}\mathbf{\mathbf{M}}$ and extend $F,V$ semi-linearly as usual. Then $\tau$ becomes $\sigma^{2}$-linear.

Let $\left\langle,\right\rangle$ be the alternating pairing on $\mathbf{M}_{k}$ satisfying

[TABLE]

This $\mathbf{M}_{k}$ is the Dieudonné module of $A_{x}[p^{\infty}]$ for any $x\in S_{ssp}(k).$ It is isomorphic111111The change in notation is made to conform with [Vo]. Previously we tried to match [Bu-We]. to the module used in part (4) of the proof of Theorem 5.2. The Lie algebra of $A_{x}$ is identified with $\mathbf{M}_{k}/p\mathbf{M}_{k}[V]=V^{-1}p\mathbf{M}_{k}/p\mathbf{M}_{k}=F\mathbf{M}_{k}/p\mathbf{M}_{k}\simeq(\mathbf{M}_{k}/V\mathbf{M}_{k})^{(p)}$ and is spanned over $k$ by $\overline{e}_{1},\overline{e}_{3},\overline{f}_{2}$.

Following [Vo] we denote $\mathbf{M}(\Sigma)=\left\langle e_{1},e_{2},e_{3}\right\rangle_{W(\kappa)}$ by $\mathbf{M}_{0}$ and $\mathbf{M}(\overline{\Sigma})$ by $\mathbf{M}_{1}$. We introduce on $\mathbf{M}_{0}$ the skew-hermitian form

[TABLE]

We extend it to a bi-additive form on $\mathbf{M}_{0,k}$ which is linear in the first variable and $\sigma$-linear in the second. It satisfies

[TABLE]

We denote the unitary isocrystal $\mathbb{Q}\otimes\mathbf{M}$ by $\mathbf{N}=\mathbf{N}_{0}\oplus\mathbf{N}_{1}$ and write also $C$ for $\mathbf{N}_{0}.$ When we base-change to the field of fractions of $W(k)$ we shall add, as before, the subscript $k$. Note that the $\mathbb{Q}_{p}$-group $\boldsymbol{J}=GU(C,\{,\})$ is isomorphic, in our case, to $\boldsymbol{G}_{/\mathbb{Q}_{p}}.$ (In general, it might be an inner form of it.)

If $\Lambda\subset C$ is a $W(\kappa)$-lattice we let

[TABLE]

If $\mathbf{M}_{k}$ were the Dieudonné module of $A_{x}[p^{\infty}]$ for a superspecial point $x,$ then the components of $S_{ss}$ passing through $x$ are classified, as we have seen before, by the set

[TABLE]

The vertices of the Bruhat-Tits tree of $\boldsymbol{J}$ are of two types. The special (s) lattices $\mathscr{L}^{(1)}$ are the lattices $\Lambda^{\prime}$ for which

[TABLE]

For example, $\mathbf{M}_{0}\in\mathscr{L}^{(1)}.$ The *hyperspecial *(hs) lattices $\mathscr{L}^{(3)}$ are those satisfying $\Lambda=\Lambda^{\vee}.$ Finally, the edges of the tree connect a lattice $\Lambda^{\prime}$ of type (s) to a vertex $\Lambda$ of type (hs) if $\Lambda^{\prime}\subset\Lambda\subset\Lambda^{\prime\vee}.$ One computes that the $p+1$ vertices of type (hs) adjacent to $\mathbf{M}_{0}$ are the lattices

[TABLE]

where $\zeta\in\mathcal{J}$ and $e_{\zeta}=p^{-1}(e_{1}+\zeta e_{3})$.

Fix $\zeta$ and let $\Lambda=\Lambda_{\zeta},$ $V=\Lambda/p\Lambda$, a vector space over $\kappa$ with basis $\overline{e}_{1},\overline{e}_{2},\overline{e}_{\zeta}.$ The skew-hermitian pairing $(,)=\{,\}\mod p$ is given in this basis by the matrix

[TABLE]

Theorems 2 and 3 of [Vo] imply the following. The $k$-points of the irreducible component of $S_{ss}$ passing through the superspecial point $x$ and labeled by $\zeta$ are in one-to-one correspondence with

[TABLE]

Here

[TABLE]

Caution has to be taken as we are over $k$ and not $\kappa:$ $(U^{\perp})^{\perp}=\tau(U)$ and not $U$. The point $x$ corresponds to $\mathbf{U}=\left\langle\overline{e}_{1},\overline{e}_{2}\right\rangle.$ In general, let $a,b\in k$ and

[TABLE]

Then

[TABLE]

is contained in $U_{a,b}$ if and only if

[TABLE]

It follows ([Vo], Lemma 4.6) that the irreducible components of $S_{ss}$ are isomorphic to the smooth projective curve whose equation is

[TABLE]

This is just the Fermat curve $x^{p+1}+y^{p+1}+z^{p+1}=0$ in disguise.

Moreover, the Dieudonné module of the abelian variety $A_{a,b}$ “sitting” at the point $(a,b)$ is

[TABLE]

where

[TABLE]

and

[TABLE]

Here $[a]$ is the Teichmüller representative of $a$ and $f_{\zeta}=p^{-1}(f_{3}-\zeta f_{1}).$

The matrices for $F$ and $V$ can now be computed. To simplify the notation let

[TABLE]

[TABLE]

Then relative to the basis $\{\epsilon_{1},\epsilon_{2},\epsilon_{3},\phi_{1},\phi_{2},\phi_{3}\}$

[TABLE]

and

[TABLE]

with

[TABLE]

6.2. The quotient $A/H$ for gss $A$

Let $(a,b)\in k^{2}$ but not in $\kappa^{2}.$ This guarantees that $A_{a,b}$ is gss, and every gss $A$ is of this sort, for an appropriate $x\in S_{ssp}(k)$ and an appropriate $\zeta\in\mathcal{J}$. Let $H\subset A[p]$ be an isotropic Raynaud subgroup scheme. Let $A^{\prime}=A/H$.

We know that $M(H)\subset M_{a,b}/pM_{a,b}$ must contain $\ker V\cap\ker F=\left\langle\overline{\epsilon}_{3}\right\rangle_{k}.$ In addition, $M(H)$ should contain a vector $\overline{\eta}$ from $M_{a,b}^{1}/pM_{a,b}^{1}$ such that $F\overline{\eta}=\overline{\epsilon}_{3}.$ We see that the most general form of such an $\eta$ is

[TABLE]

$(u:v)\in\mathbb{P}^{1}(W(k)).$ Thus

[TABLE]

Note that by the assumption that $(a,b)$ is not in $\kappa^{2},$ neither $a$ nor $b$ lies in $\kappa.$ Thus $H$ is uniquely classified by $(\overline{u}:\overline{v})\in\mathbb{P}^{1}(k)$. The point $\overline{u}=0$ corresponds to an $H$ such that $M(H)$ is killed by $F$, or $H$ is killed by $Ver$. This $H$ will be of type $\alpha_{p^{2},\Sigma}$ and $(\underline{A},H)$ will lie then on $Z_{et}.$ The point $\overline{v}=0$ will correspond to an $H$ such that $M(H)$ is killed by $V,$ or $H$ is killed by $Frob.$ This $H$ will be of type $\alpha_{p^{2},\Sigma}^{*}$ and $\left(\underline{A},H\right)$ will lie then on $Z_{m}.$

Assume from now on that we are not in these two special cases, so that $H$ is of type $\mathfrak{G}_{\Sigma}[p].$ Then $M(A^{\prime}[p^{\infty}])=M^{\prime}$ will sit in an exact sequence

[TABLE]

and inside $\mathbf{N}_{k}$, $M^{\prime}=\left\langle\epsilon_{1},\epsilon_{2},p^{-1}\epsilon_{3},\phi_{1},p^{-1}\eta,\phi_{3}\right\rangle_{W(k)},$ provided $\overline{u}\neq-\overline{v}$. If $\overline{u}=-\overline{v}$ the same basis works, if we replace $\phi_{3}$ by $\phi_{2}$. Assume from now on that $\overline{u}\neq-\overline{v}$. We calculate the matrices of $F$ and $V$ in this basis as we did for $M=M_{a,b}$ before. The matrix of $F$ comes out to be

[TABLE]

where we put $w=(u[b^{p}]+v[b^{1/p}])(u+v)^{-1},$ while the one of $V$ is

[TABLE]

We see that $M^{\prime}/pM^{\prime}[V]\cap M^{\prime}/pM^{\prime}[F]$ is spanned by the images modulo $pM^{\prime}$ of $\phi_{3}$ and of $x\epsilon_{1}+y\epsilon_{2}+zp^{-1}\epsilon_{3}$ provided $x,y,z\in W(k)$ are such that

[TABLE]

[TABLE]

These two equations are equivalent to

[TABLE]

[TABLE]

The solution set $(x,y,z)\mod p$ to these two equations is 1-dimensional, unless $\overline{w}\in\kappa$ and $\overline{b}^{p-1/p}=1$, where it is 2-dimensional. This last condition however translates into $\overline{b}\in\kappa$, which we assumed not to be the case. We conclude that $M^{\prime}/pM^{\prime}[V]\cap M^{\prime}/pM^{\prime}[F]$ is always two-dimensional, of type $(\Sigma,\overline{\Sigma}).$ This settles the $a$-type of $A^{\prime}$ in the cases that were deferred to the appendix in the proof of Theorem 5.2.