Categoricity of Shimura Varieties

TL;DR

This paper develops a model-theoretic framework for Shimura varieties, establishing conditions for categoricity linked to number theory conjectures, and proves categoricity for specific cases of moduli spaces of abelian varieties.

Contribution

It introduces necessary and sufficient model-theoretic conditions for Shimura varieties to be categorical, connecting these to key number theory conjectures.

Findings

Categoricity conditions are linked to Galois representations.

Proves categoricity for f6A_2 and f6A_3.

Existing literature suffices for these proofs.

Abstract

We propose a model-theoretic structure for Shimura varieties and give necessary and sufficient conditions to obtain categoricity. We show that these conditions are directly related to important conjectures in number theory coming from Galois representations attached the points of a Shimura variety. We end by showing that the existing literature is enough to prove categoricity of $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$.

Equations

$$\begin{array}{ccc}{X}^{+}& \stackrel{\xi }{\to }& X^{+}\\ \downarrow p& & \downarrow q\\ S(\mathbb{C})& \stackrel{\sigma }{\to }& S(\mathbb{C}),\end{array}$$

$$M_{\left\{p\right\}}\text{vbox}\text{oalign}\text{cr}\text{hfil}|\text{hfil}\text{cr}\text{cr}⌣\text{cr}\text{cr}{\text{displaylimits}}_{M_{\varnothing }}M\ngeqq \left\{p\right\}.$$

$$\Gamma (N):=\left\{g\in G(\mathbb{Z}):g\equiv \mathrm{id}\text{mod}N\right\}.$$

$${\mathrm{Sh}}_K(G,X):=G(\mathbb{Q})\backslash \left(X\times \left(G\left({\mathbb{A}}_f\right)/K\right)\right),$$

$$\coprod _{g\in \mathcal{C}}{\Gamma }_g\backslash X^{+}\cong G(\mathbb{Q})\backslash \left(X\times \left(G\left({\mathbb{A}}_f\right)/K\right)\right)$$

$${\Gamma }_g\backslash X^{+}$$

$$[x{]}_{{\Gamma }_g}$$

$$a+ib\mapsto \left(\begin{array}{cc}a& -b\\ b& a\end{array}\right).$$

$$J_g:=\left(\begin{array}{cc}0& I_g\\ -I_g& 0\end{array}\right),$$

$${\mathrm{GSp}}_{2g}(\mathbb{R}):=\left\{M\in {\mathrm{GL}}_{2g}(\mathbb{R}):M^t{J}_{g}M=v(M)J_g\right\},$$

$${\mathrm{Sp}}_{2g}:=\left\{M\in {\mathrm{GL}}_{2g}:M^t{J}_{g}M=J_g\right\}.$$

$$h:$$

$$a+ib\mapsto a{I}_{2g}+b{J}_g$$

$${\mathbb{H}}_g^{+}:=\left\{Z=A+iB\in M_{g\times g}(\mathbb{C}):Z=Z^t,B>0\right\}$$

$$\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\cdot Z:=(AZ+B)(CZ+D{)}^{-1}.$$

$${\pi }_{{\Gamma }_i}:{\mathbb{X}}^{+}$$

$${\left({\left[x_i\right]}_{{\Gamma }_i}\right)}_{{\Gamma }_i}$$

$${\rho }_{[y{]}_{{\Gamma }^{\prime }}}:\mathrm{Gal}\left(\overline{L}/L\right)\to \Gamma /{\Gamma }^{\prime }.$$

$$\mathrm{Sh}(G,X):=G(\mathbb{Q})\backslash \left(X\times G({\mathbb{A}}_f)\right),$$

$${\pi }_K:\mathrm{Sh}(G,X)$$

$$[x,g]$$

$${\rho }_{\tilde{y}}:\mathrm{Gal}\left(\overline{L}/L\right)\to \hat{\Gamma }.$$

$$f:X_{V,i}^{+}$$

$$(x_1,\dots ,x_m)$$

$$\varphi :H$$

$$\left(h_1,\dots ,h_m\right)$$

$$Z_{\overline{g}}^{V,i}:=\left\{\left(p\left(\overline{g}{x}_1\right),\dots ,p\left(\overline{g}{x}_m\right)\right):\left(x_1,\dots ,x_m\right)\in X_{V,i}^{+}\right\}.$$

$${\Gamma }_{\overline{g}}:=\Gamma \cap g_1^{-1}\Gamma g_1\cap \cdots \cap g_n^{-1}\Gamma g_n.$$

$${\psi }_{\overline{h},\overline{g}}:Z_{\overline{h}}\to Z_{\overline{g}}$$

$$\mathrm{Aut}(\mathbb{C}/L)\to {\Gamma }_{\overline{g}}/{\Gamma }_{\overline{h}}.$$

$${\psi }_{\overline{h},\overline{g}}^{V,i}:Z_{\overline{h}}^{V,i}\to Z_{\overline{g}}^{V,i}.$$

$$\overline{\Gamma }:=\underset{\overline{g}}{\underleftarrow{\lim }}\Gamma /{\Gamma }_{\overline{g}}\text{mbox}and\overline{{\Gamma }^{V,i}}:=\underset{\overline{g}}{\underleftarrow{\lim }}{\Gamma }_{e,\overline{g}}^{V,i},$$

$${\Gamma }^{\prime \prime }:=\bigcap _{\gamma \in \Gamma }\gamma {\Gamma }^{\prime }{\gamma }^{-1}$$

Full text

1. 1 Introduction

2. 2 Model Theory

3. 2.1 Quantifier Elimination

4. 2.2 Pregeometries

5. 2.3 Quasiminimality

6. 2.4 $\ell$-Isolation

7. 2.5 Independent Systems

8. 3 Introduction to Shimura Varieties

9. 3.1 Congruence Subgroups

10. 3.2 Connected Shimura Varieties

11. 3.3 Classifying Points

12. 3.4 Special Subvarieties

13. 3.5 Galois Representations I

14. 3.6 Galois Representations II

15. 3.7 Normal Subgroups in $\Gamma$

16. 4 Shimura Structures

17. 4.1 Theory of the Variety Sort

18. 4.2 Axiomatisation

19. 4.3 Quantifier Elimination

20. 4.4 Special Locus

21. 5 Models With Standard Fibres

22. 6 Shimura Covers

23. 6.1 The covering sort

24. 6.2 Axiomatisation and quantifier elimination

25. 6.3 Stability Aspects

26. 7 Necessary Conditions for Categoricity

27. 8 Categoricity Via Quasiminimality

28. 8.1 Hecke-generic Curves

29. 8.2 Finite-Index Conditions

30. 8.3 Conclusions

31. 9 Classification of Shimura Covers

32. 9.1 Preliminaries

33. 9.2 $\omega$-Stability Over Models

34. 9.3 Atomicity Over Independent Systems

35. 9.4 Classification of Models with Standard Fibres

36. 10 Additional Remarks

37. 10.1 Reduction to the Neat Case

38. 10.2 Categoricity of $\mathcal{A}_{g}$

1. Introduction

Suppose that $p:X^{+}\rightarrow S(\mathbb{C})$ is a (connected) Shimura variety given by the Shimura datum $(G,X)$, that $E^{S}$ is the minimal abelian extension of the reflex field $E(G,X)$ over which $S$ is defined, and let $\Sigma$ be the set of coordinates of special points in $S(\mathbb{C})$. Briefly speaking then, the categoricity problem for $S$ is: if $q:X^{+}\rightarrow S(\mathbb{C})$ is any other map (of sets) satisfying the same algebraic conditions as $p$ (this will be made precise later: we will ask that $q$ satisfy the Special Subvarieties and Standard Fibers conditions), is there then a commutative diagram:

[TABLE]

where $\xi:X^{+}\rightarrow X^{+}$ is a $G^{\mathrm{ad}}(\mathbb{Q})^{+}$-equivariant bijection which respects special domains (see §3.4), and $\sigma$ is a map induced by an automorphism of $\mathbb{C}$ fixing $E^{S}(\Sigma)$?

Another way of thinking about categoricity in this case is: how rigid is the special structure of $S$? Or, to what extent is $p$ determined by the algebraic properties of special subvarieties of $S$? A precise definition of categoricity is given at the beginning of §2.

Categoricity for Shimura curves was established in [10], and the authors acknowledged in their paper that the language they used may not be rich enough to treat the higher dimensional cases. The first objective of the present work then is to provide an enriched language which keeps the nice properties of the language used in the one-dimensional case: quantifier elimination, a useful description of types, and superstability. The second objective is to give necessary and sufficient conditions for Shimura varieties of any dimension to be categorical. It is immediate from the work done in [10] (and from other similar categoricity results which will be addressed later in this Introduction) that some form of arithmetic open image condition is needed. Our main goal is to postulate the correct open image condition (here we split it into two conditions: FIC1 and FIC2) and prove that it is equivalent to categoricity. This open image condition is known to hold in dimension $1$ thanks to [10, Theorem 1.1], but not known in general. In fact, a version of this condition had already been conjectured by Pink in [20] (see Conjecture 1), and he also shows the link between his conjecture and the Mumford-Tate conjecture. We will be able, however, to show that enough results are known in the literature to obtain categoricity of two well-known Shimura varieties: $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$, the moduli spaces of principally polarised abelian varieties of dimension 2 and 3, respectively. As far as we know, this result is new.

The categoricity problem will be approached adapting techniques on $\ell$-atomicity and independent systems developed in [4] for studying commutative groups of finite Morley rank (e.g. abelian varieties). We will also show an approach using quasiminimal classes that gives the same result, but we will prefer the first method as it proves to be more robust than the quasiminimal approach. We show both approaches as they emphasize different aspects of the problem, the methods of [4] focusing more on the model-theoretic properties of the structures, while the quasiminimal approach focuses on the existence of certain curves within the Shimura variety.

After going over some necessary concepts and results from model-theory in §2, we give an introduction to Shimura varieties in §3. In §4, §5 and §6 we present the model-theoretic structures for Shimura varieties, we define the special subvarieties condition (here called $\mathrm{SS}$), we prove quantifier elimination of this language and give an axiomatisation for the complete first-order theory of $p:X^{+}\rightarrow S(\mathbb{C})$. We also give a description of the types in models of our theory that satisfy the standard fibres condition (here called SF). Using what we will call Shimura covers, we prove some stability properties of the first-order theory of a Shimura variety.

In §8 and §9 we present our main categoricity results, first using quasiminimal classes, and then through independent systems. At the end of §10 we show that the categoricity conditions FIC 1 & 2 for $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$ can be obtained from the literature.

This work follows an important series of results regarding categoricity of certain algebraic varieties arising from arithmetic: [10] for categoricity of Shimura curves, [4] for categoricity of abelian and split semi-abelian varieties, [5] for categoricity conditions of pseudo-exponential maps, [28] for the general quasiminimal strategy for semi-abelian varieties, [6] and [29] for categoricity of multiplicative groups of algebraically closed fields, [14] for categoricity of abelian varieties under continuum hypothesis, [2] for categoricity of elliptic curves without complex multiplication; among others.

2. Model Theory

In this section we give some standard definitions and results from model theory needed for our work. We will always assume that the language $\mathcal{L}$ we are using is countable. For us the word type means complete type. Standard definitions and results from model theory can be found, for example, in [24].

Definition**.**

Let $\kappa$ be an infinite cardinal. An $\mathcal{L}$-theory $T$ is $\kappa$-categorical if all models of $T$ of cardinality $\kappa$ are isomorphic. We will say that $T$ is categorical if it is $\kappa$-categorical for all infinite $\kappa$.

Example 1**.**

Consider the theory of algebraically closed fields: ACF. We use $\mathrm{ACF}_{0}$ to denote the theory of algebraically closed fields of characteristic 0. Then $\mathrm{ACF}_{0}$ is $\kappa$-categorical for any uncountable $\kappa$, but it is not $\aleph_{0}$-categorical, as $\overline{\mathbb{Q}}$ and $\overline{\mathbb{Q}(X)}$ are non-isomorphic countable models of $\mathrm{ACF}_{0}$.

Proposition 2.1** (see [24, §3.3.4]).**

Let $K\models\mathrm{ACF}$ and $A\subseteq K$. Let $k$ be the subfield of $K$ generated by $A$. Then the $n$-types over $A$ are determined by prime ideals in the poynomial ring $k[X_{1},\ldots,X_{n}]$.

This means that if $m_{1},\ldots,m_{n}\in K$, then $\mathrm{tp}(m_{1},\ldots,m_{n}/A)$ is determined by the smallest algebraic subvariety $W\subseteq K^{n}$ defined over $k$ which contains the point $(m_{1},\ldots,m_{n})$. Any other point of $W$ which is generic in $W$ over $k$, will also realise the type $\mathrm{tp}(m_{1},\ldots,m_{n}/A)$.

Definition**.**

An $\mathcal{L}$-structure $M$ is strongly minimal if every definable subset of $M$ is either finite or cofinite.

Example 2**.**

$\mathrm{ACF}_{0}$ is strongly minimal (see [24, §5.7]).

2.1. Quantifier Elimination

We recall a few standard definitions and results about theories with quantifier elimination. The exposition is taken from [19].

Definition**.**

Let $M$ and $N$ be $\mathcal{L}$-structures. A finite partial isomorphism between $M$ and $N$ is a pair $\left(\overline{a},\overline{b}\right)$ with $\overline{a}\in M$ and $\overline{b}\in N$, such that $\mathrm{qftp}_{M}\left(\overline{a}\right)=\mathrm{qftp}_{N}\left(\overline{b}\right)$.

In other words, a finite partial isomorphism between $M$ and $N$ is the same thing as an isomorphism of $\mathcal{L}$-structures between a finitely generated substructure of $M$ and a finitely generated substructure of $N$.

Definition**.**

Let $M$ and $N$ be $\mathcal{L}$-structures. Let $I$ be a set of partial isomorphisms between $M$ and $N$. $I$ has back-and-forth if for every partial isomorphism $\left(\overline{a},\overline{b}\right)\in I$ and every $c\in M$, there is $d\in N$ such that $\left(\overline{a}c,\overline{b}d\right)\in I$ and dually.

Definition**.**

An $\mathcal{L}$-structure $M$ is $\aleph_{0}$-homogeneous if for every finite subset $A\subseteq M$ and very $m\in M$, any elementary map $A\rightarrow M$ can be extended to an elementary map $A\cup\left\{m\right\}\rightarrow M$.

When talking about the class of models of a theory $T$, the previous concept will appear in the following two variants:

1. (1)

$\aleph_{0}$-homogeneity over the empty set. This will mean that, given any two models of $T$, the set $I$ of all finite partial isomorphism between them has back-and-forth. 2. (2)

$\aleph_{0}$-homogeneity over countable models. For this we need to adapt the definition of finite partial isomorphism. Given $M,M^{\prime},N\models T$, where $N$ is a countable submodel common to $M$ and $M^{\prime}$, a partial isomorphism over $N$ is a pair $\left(\overline{a},\overline{b}\right)$ with $\overline{a}\in M$ and $\overline{b}\in M^{\prime}$, such that $\mathrm{qftp}_{M}\left(\overline{a}/N\right)=\mathrm{qftp}_{M^{\prime}}\left(\overline{b}/N\right)$. Now we can define $\aleph_{0}$-homogeneity over countable models in an analogous way to $\aleph_{0}$-homogeneity over the empty set.

The following proposition gives us the strategy we will use later for proving quantifier elimination for Shimura structures.

Proposition 2.2** (see [19, Proposition 2.29]).**

Let $T$ be an $\mathcal{L}$-theory. The following are equivalent:

1. (a)

$T$* has quantifier elimination.* 2. (b)

If $M$ and $N$ are $\omega$-saturated models of $T$ and $I$ is the set of all finite partial isomorphism between $M$ and $N$, then $I$ has back-and-forth.

The same strategy can also be used to prove completeness of the theory.

Proposition 2.3** (see [19, Proposition 2.30]).**

Suppose that for any two $\omega$-saturated models $M$ and $N$ of $T$, the set of finite partial isomorphisms between $M$ and $N$ has back-and-forth and is non-empty. Then $T$ is complete.

2.2. Pregeometries

Definition**.**

Let $X$ be a set. A function $\mathrm{cl}:\mathcal{P}(X)\rightarrow\mathcal{P}(X)$ (here $\mathcal{P}(X)$ denotes the power set of $X$) is called a pregeometry on $X$ if it satisfies the following properties for every $A,B\in\mathcal{P}(X)$:

1. (a)

$A\subseteq\mathrm{cl}(A)$, 2. (b)

If $A\subseteq B$, then $\mathrm{cl}(A)\subseteq\mathrm{cl}(B)$, 3. (c)

$\mathrm{cl}(\mathrm{cl}(A))=\mathrm{cl}(A)$, 4. (d)

Finite character: if $a\in\mathrm{cl}(A)$, then there is a finite subset $A_{0}\subseteq A$ such that $a\in\mathrm{cl}(A_{0})$, 5. (e)

Exchange: if $a,b\in X$ are such that $a\in\mathrm{cl}(A\cup\left\{b\right\})$ and $a\notin\mathrm{cl}(A)$, then $b\in\mathrm{cl}(A\cup\left\{a\right\})$.

If $\mathrm{cl}$ is a pregeometry on $X$, then a subset $A\subseteq X$ is called $\mathrm{cl}$-closed if $\mathrm{cl}(A)=A$.

A crucial aspect of pregeometries is that they allow us to have well-defined notions of independence and dimension. Suppose $\mathrm{cl}$ is a pregeometry on $X$. A set $A\subseteq X$ is $\mathrm{cl}$-independent if for every $a\in A$ we have that $a\notin\mathrm{cl}(A\setminus\left\{a\right\})$. The dimension of a set $B\subseteq X$ with respect to $\mathrm{cl}$ is the cardinality of any $\mathrm{cl}$-independent set $A\subseteq B$ such that $\mathrm{cl}(A)=B$. For more basic properties of pregeometries, see [24, Appendix C].

Example 3** (Some pregeometries).**

We give two well-known examples of pregeometries and the dimensions they define.

1. (1)

If $X$ is a $K$-vector space, we can take $\mathrm{cl}(A):=\mathrm{lin.span}_{K}(A)$. In this case $\dim A=\mathrm{lin.dim.}_{K}(\mathrm{cl}(A))$. 2. (2)

If $X$ is a field, we can take $\mathrm{cl}(A)$ to be the set of elements of $X$ which are algebraic over $A$. In this case $\dim A=\mathrm{tr.deg.}(A)$. This pregeometry is usually denoted (and this is the name we will use): $\mathrm{acl}$. This example is an instance of a more general result, namely that the model-theoretic notion of algebraic closure defines a pregeometry on strongly minimal sets (see [24, Theorem 5.7.5]).

2.3. Quasiminimality

In this subsection we introduce the terminology related to quasiminimality and quasiminimal structures as presented in [3] and [16]. The final theorem of this subsection provides the strategy we will follow to obtain one of our main results: Theorem 8.7. As always, we assume $\mathcal{L}$ is countable.

Definition**.**

An $\mathcal{L}$-structure $M$ is quasiminimal is every definable subset of $M$ is countable or co-countable.

Definition**.**

Let $M$ be an $\mathcal{L}$-structure. Let $\mathrm{cl}$ be a pregeometry on $M$. Then the pair $(M,\mathrm{cl})$ is called a quasiminimal pregeometry structure if:

**(QM1): **

$\mathrm{cl}$ is determined by the language: given $a,a^{\prime},b_{1},\ldots,b_{n},b_{1}^{\prime},\ldots,b_{n},\in M$ such that $\mathrm{tp}\left(a,\overline{b}\right)=\mathrm{tp}\left(a^{\prime},\overline{b}^{\prime}\right)$, then $a\in\mathrm{cl}\left(\overline{b}\right)$ if and only if $a^{\prime}\in\mathrm{cl}\left(\overline{b}^{\prime}\right)$.

**(QM2): **

$M$ is infinite dimensional with respect to $\mathrm{cl}$.

**(QM3): **

Countable Closure Property. If $A\subseteq M$ is finite, then $\mathrm{cl}(A)$ is countable.

**(QM4): **

Uniqueness of the generic type. Suppose $C,C^{\prime}\subseteq M$ are countable closed subsets, enumerated such that $\mathrm{tp}(C)=\mathrm{tp}(C^{\prime})$. If $a\in M\setminus C$ and $a^{\prime}\in M\setminus C^{\prime}$, then $\mathrm{tp}(C,a)=\mathrm{tp}\left(C^{\prime},a^{\prime}\right)$ (with respect to the same enumerations for $C$ and $C^{\prime}$).

**(QM5): **

$\aleph_{0}$-homogeneity over closed sets and the empty set. Let $C,C^{\prime}\subseteq M$ be countable closed subsets or empty, enumerated such that $\mathrm{tp}(C)=\mathrm{tp}\left(C^{\prime}\right)$, and let $\overline{b},\overline{b}^{\prime}$ be finite tuples from $M$ such that $\mathrm{tp}\left(C,\overline{b}\right)=\mathrm{tp}\left(C^{\prime},\overline{b}^{\prime}\right)$, and let $a\in\mathrm{cl}\left(H,\overline{b}\right)$. Then there is $a^{\prime}\in M$ such that $\mathrm{tp}\left(C,\overline{b},a\right)=\mathrm{tp}\left(C,^{\prime}\overline{b}^{\prime},a^{\prime}\right)$.

$(M,\mathrm{cl})$ is said to be a weakly quasiminimal pregeometry structure if it satisfies all the axioms except maybe (QM2).

Definition**.**

Let $(M,\mathrm{cl})$ be a quasiminimal pregeometry structure. Let $\mathscr{K}^{-}(M)$ be the smallest class of $\mathcal{L}$-structures that contains $M$, all the $\mathrm{cl}$-closed substructures of $M$, and that is closed under isomorphism. Let $\mathscr{K}(M)$ be the smallest class containing $\mathscr{K}^{-}(M)$ and that is closed under unions of chains of closed embeddings. Any class of structures of the form $\mathscr{K}(M)$ is called a quasiminimal class.

Now we present the main result about categoricity and quasiminimal classes.

Theorem 2.4** (see [3, Theorem 2.3]).**

If $\mathscr{K}$ is a quasiminimal class, then every structure $A\in\mathscr{K}$ is a weakly quasiminimal pregeometry structure and, up to isomorphism, there is exactly one structure in $\mathscr{K}$ of each cardinal dimension. In particular, $\mathscr{K}$ is uncountably categorical. Furthermore, $\mathscr{K}$ is the class of models of an $\mathcal{L}_{\omega_{1},\omega}$-sentence.

It will be important for us that this theorem can be applied to structures defined by $\mathcal{L}_{\omega_{1},\omega}$-sentences (which are sentences that can be infinitely long, with countably many conjunctions and disjunctions, but with only finitely many variables) because the Standard Fibres condition (see Subection 5) will be such a sentence.

Proposition 2.5** (see [16, Corollary 2.2]).**

Let $\mathscr{K}$ be a quasiminimal class. Then models of $\mathscr{K}$ of dimension less than or equal to $\aleph_{1}$ are determined up to isomorphism by their dimension. There is at most one model of cardinality $\aleph_{1}$.

2.4. $\ell$-Isolation

The definitions and results of this subsection can be found in [4, §3].

Definition**.**

Given a type $p\in S(B)$ and a formula $\phi(x,y)$, the complete $\phi$-type implied by $p$, denoted $p_{\phi}$, is the set of all formulas of the form $\phi\left(x,\overline{b}\right)$ that are in $p$, where $\overline{b}$ is a tuple from $B$.

A type $p$ is $\ell$-isolated (which is short for locally isolated) if for each $\phi(x,y)$, there exists $\psi(x)\in p$ such that $\psi$ implies the complete $\phi$-type implied by p, i.e. $\psi\models p_{\phi}$.

$A$ is said to be atomic over $B$ if $\mathrm{tp}(a/B)$ is isolated for each tuple $a\in A$. $A$ is constructible over $B$ if $A$ has an enumeration $(a_{i})_{i\leq\lambda}$ such that $\mathrm{tp}\left(a_{i}/Ba_{\leq i}\right)$ is isolated for each $i\leq\lambda$, where $Ba_{\leq i}=B\cup\left\{a_{j}:j\leq i\right\}$.

We say $A$ is $\ell$-atomic and $\ell$-constructible over $B$ by replacing isolation in the definitions above with $\ell$-isolation.

Remark 2.6*.*

It is easy to see that isolated types are $\ell$-isolated, so atomicity implies $\ell$-atomicity and constructibility implies $\ell$-constructibility. Also constructibility implies atomicity and $\ell$-constructibility implies $\ell$-atomicity ([23, Theorem IV.3.2]). The converse statements hold for countable sets ([23, Lemma IV.3.16]).

Lemma 2.7** (see [4, Lemma 3.6-a)]).**

Let $\mathbb{M}$ be a monster model of a complete stable theory $T$.

1. (a)

For $A\subseteq\mathbb{M}^{\mathrm{eq}}$, there exists $M\prec\mathbb{M}$ such that $A\subseteq M^{\mathrm{eq}}$ and $M^{\mathrm{eq}}$ is $\ell$-constructible over $A$. 2. (b)

Let $M\prec\mathbb{M}$. If $\phi$ is a formula over $M$ such that $\phi(M)\subseteq A\subseteq\mathbb{M}^{\mathrm{eq}}$ and $\mathrm{dcl}^{\mathrm{eq}}(A)\cap\mathrm{dcl}^{\mathrm{eq}}(\phi(\mathbb{M}))\subseteq M^{\mathrm{eq}}$, and if $b$ is $\ell$-isolated over $A$ and $\models\phi(b)$, then $b\in\phi(M)$.

2.5. Independent Systems

Here we give some definitions and results from [4, §3.5] necessary for the classification of Shimura covers in Section 6. Let $T$ be a stable theory and $\mathbb{M}$ be a monster model of $T$.

Definition**.**

Let $I$ be a downward-closed set of sets. An $I$-system in $\mathbb{M}$ is a collection $\left(M_{s}\right)_{s\in I}$ of elementary submodels of $\mathbb{M}$ such that for $s\subseteq t$, $M_{s}$ is an elementary submodel of $M_{t}$. Given $J\subseteq I$, let $M_{J}:=\bigcup_{s\in J}M_{s}\subseteq\mathbb{M}$.

For $s\in I$, define $<s:=\mathcal{P}(s)\setminus\left\{s\right\}$ and $\ngeq s:=I\setminus\left\{t:s\subseteq t\right\}$.

The $I$-system $\left(M_{s}\right)_{s\in I}$ is called constructible (resp. atomic, $\ell$-constructible, $\ell$-atomic) if $M_{s}$ is constructible (resp. atomic, $\ell$-constructible, $\ell$-atomic) over $M_{<s}$ for all $s\in I$ with $|s|>1$.

The $I$-system is called independent (or stable) if M_{s}\mathrel{\mathop{\vbox{ \hbox{\oalign{\kern-1.29167pt\cr\hfil|$\hfil\cr\kern-3.01385pt\cr$\smile\cr\kern-1.29167pt\cr}} }}\displaylimits_{M_{<s}}}M_{\ngeq s} for all $s\in I$.

The $I$-system is Noetherian if each $s\in I$ is finite.

An enumeration of $I$ is a sequence $(s_{i})_{i\leq\lambda}$, where $\lambda$ is an ordinal, such that $I=\left\{s_{i}:i\in\lambda\right\}$ and $s_{i}\subseteq s_{j}\implies i\leq j$. We write $s_{<i}$ for $\left\{s_{j}:j<i\right\}$.

Lemma 2.8** (see [23, Lemma XII.2.3(1)]).**

Let $(M_{s})_{s}$ be an $I$-system, and let $(s_{i})_{i\in\lambda}$ be an enumeration. Then the system is independent if and only if M_{s_{i}}\mathrel{\mathop{\vbox{ \hbox{\oalign{\kern-1.29167pt\cr\hfil|$\hfil\cr\kern-3.01385pt\cr$\smile\cr\kern-1.29167pt\cr}} }}\displaylimits_{M_{<s_{i}}}}M_{s_{<i}} holds for all $i\in\lambda$.

Lemma 2.9** (see [4, Lemma 3.21]).**

Let $(M_{s})_{s}$ be a constructible Noetherian independent $I$-system. Suppose that for each $p\in\bigcup I$, $B_{p}$ is a subset of $M_{\left\{p\right\}}$ for which $M_{\left\{p\right\}}$ is constructible over $M_{\emptyset}B_{p}$. Then $M_{I}$ is constructible over $A:=M_{\emptyset}\cup\bigcup_{p\in\bigcup I}B_{p}$.

Lemma 2.10** (see [4, Lemma 3.22]).**

Suppose $\bigcup I$ is finite. For an $\ell$-atomic $I$-system $(M_{s})_{s}$ to be independent, it suffices to show that for each $p\in\bigcup I$

[TABLE]

3. Introduction to Shimura Varieties

A standard introduction to this subject is [18], which is a very complete treatment of Shimura varieties. On the other hand, as we will only be dealing with connected varieties, [20] is also a very good starting point. We will use both sources in this work. An important observation though is that in [20] the author treats “connected mixed Shimura varieties”, whereas [18] mainly refers to “pure Shimura varieties”. In our work we will work with pure Shimura varieties, only because there are a few results which hold for the pure case for which we could not find a reference in the mixed case. We expect that the methods of this paper can be generalised to the mixed case. Whenever possible, we will point out if a certain result can be easily extended from the pure case to the mixed case.

3.1. Congruence Subgroups

Let $G\subseteq\mathrm{GL}_{n}(\mathbb{C})$ be a linear algebraic group over $\mathbb{Q}$ and let $G(\mathbb{Z})$ denote $G(\mathbb{Q})\cap\mathrm{GL}_{n}(\mathbb{Z})$.

Definition**.**

A subgroup $\Gamma$ of $G(\mathbb{Q})$ is arithmetic if $\Gamma$ has finite index in both $\Gamma$ and $G(\mathbb{Z})$, in other words, $\Gamma$ and $G(\mathbb{Z})$ are commensurable.

Next we define a distinguished family of arithmetic subgroups of $G(\mathbb{Z})$.

Definition**.**

Given an embedding of $G$ into $\mathrm{GL}_{n}$, the principal congruence subgroup of level $N$ in $G$ is:

[TABLE]

This definition depends on the embedding. To address this issue, we define a congruence subgroup of $G$ to be a subgroup $\Gamma$ of $G$ such that for some $N$, $\Gamma(N)$ is a finite-index subgroup of $\Gamma$. This definition does not depend on the embedding.

Definition**.**

An element $g\in G(\mathbb{Q})$ is called neat if for some (for all) faithful representation $\rho:G\hookrightarrow\mathrm{GL}(V)$, the eigenvalues of $\rho(g)$ in $\mathbb{C}$ generate a torsion-free subgroup of $\mathbb{C}^{\times}$. A subgroup of $G(\mathbb{Q})$ is neat, if all its elements are neat.

Proposition 3.1** (see [18, Proposition 3.5]).**

Let $G$ be an algebraic group over $\mathbb{Q}$, and let $\Gamma$ be an arithmetic subgroup of $G(\mathbb{Q})$. Then $\Gamma$ contains a neat subgroup $\Gamma^{\prime}$ of finite index. Moreover, $\Gamma^{\prime}$ can be defined by congruence relations (i.e. for some embedding $G\hookrightarrow\mathrm{GL}_{n}$ and integer $N$, $\Gamma^{\prime}=\left\{g\in\Gamma:g\equiv\mathrm{id}\,\mathrm{mod}\,N\right\}$).

3.2. Connected Shimura Varieties

Let $\mathbb{S}$ be the Deligne torus, that is, the linear algebraic group over $\mathbb{R}$ such that $\mathbb{S}(\mathbb{R})=\mathbb{C}^{\times}$.

A Shimura datum is a pair $(G,X)$ consisting of a linear algebraic group $G$ over $\mathbb{Q}$ and the $G(\mathbb{R})$-conjugacy class $X$ of a morphism $h:\mathbb{S}\rightarrow G_{\mathbb{R}}$ satisfying some conditions on $G$, the complex structure of $X$, and the action of $G$ on $X$: see axioms SV1-SV3 in [18, p. 302] for the “pure” axioms, or [20, Definition 2.1] for the “mixed” ones. More important than the precise statement of these axioms is that they guarantee that quotients of connected components of $X$ by certain arithmetic subgroups will give an algebraic variety. For this work, we will use the definitions and notation of [18] for pure Shimura varieties.

Remark 3.2*.*

The fact that we do not need the precise statement of axioms of Shimura varieties could be interpreted as saying that the work we will do here may be replicated on other algebraic varieties obtained as the quotient of some complex domain by the action of a discrete group. However, as it will become apparent, what really makes the model-theory work is the nice behaviour of the geometry of the “special” subvarieties (see §3.4). Therefore, the present work, as well as [4] and [10], should be seen as giving insight to the essential properties of the “special structure” of these algebraic varieties. Finding an abstract definition of “special structure” is an important question in Diophantine geometry and some attempts at this definition have been made, see for instance [25].

Let $G^{\mathrm{ad}}$ denote $G$ modulo its centre, let $G^{\mathrm{ad}}(\mathbb{R})^{+}$ be the connected component of the identity in $G^{\mathrm{ad}}(\mathbb{R})$, and let $G^{\mathrm{ad}}(\mathbb{Q})^{+}:=G^{\mathrm{ad}}(\mathbb{Q})\cap G^{\mathrm{ad}}(\mathbb{R})^{+}$. Let $X^{+}$ denote a connected component of $X$; its stabiliser in $G^{\mathrm{ad}}(\mathbb{R})$ is $G^{\mathrm{ad}}(\mathbb{R})^{+}$. We call $(G,X^{+})$ a connected Shimura datum.

Let $K$ be a compact open subgroup of $G\left(\mathbb{A}_{f}\right)$ (where $\mathbb{A}_{f}$ denotes the finite rational adèles). Define the double coset space:

[TABLE]

where $G(\mathbb{Q})$ acts diagonally. We will denote the elements of this space by $[x,gK]$. The following lemma will show that $\mathrm{Sh}_{K}(G,X)$ can be seen as a finite disjoint union of connected components. To that end, let $G(\mathbb{Q})_{+}$ denote the stabiliser of $X^{+}$ in $G(\mathbb{Q})$, and let $\mathscr{C}$ be a set of representatives for the finite set $G(\mathbb{Q})_{+}\backslash G(\mathbb{A}_{f})/K$. Given $g\in\mathscr{C}$, let $\Gamma_{g}:=G(\mathbb{Q})_{+}\cap gKg^{-1}$ (which is a congruence subgroup of $G$), and let $[x]_{\Gamma_{g}}$ denote the class of an element $x\in X^{+}$ in the quotient $\Gamma_{g}\backslash X^{+}$.

Lemma 3.3** (see [18, Lemma 5.13]).**

If $X^{+}$ is a connected component of $X$ and we endow $G(\mathbb{A}_{f})$ with the adèlic topology, then there is a homeomorphism

[TABLE]

which is given by

[TABLE]

The axioms of Shimura data ensure that, when the $\Gamma_{g}$ are sufficiently small, the members of the disjoint union in (3.2) have canonical realisations as quasi-projective varieties over $\mathbb{C}$. In fact, by [18, Theorem 3.12] and [20, Fact 2.3], the $\Gamma_{g}$ are sufficiently small when their images in $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ are neat. Unless stated otherwise, we will assume from now on that $\Gamma$ is neat.

Remark 3.4*.*

Even though we assume $\Gamma$ to be neat, a straight-forward generalisation of [10, §5.3] shows that one can obtain a commutative diagram like (1.1) for the general case from the neat case. This is done in §10.1.

$\mathrm{Sh}_{K}(G,X)$ possesses a canonical model over its reflex field $E:=E(G,X)$ which is independent of the choice of $K$ (see [18, §14] for an exposition and references). For a precise definition of the reflex field, look at [18, §12]. We point out that $E$ is a number field (see [18, Remark 12.3-a)]).

In the notation of (3.1), let $\Gamma=\Gamma_{e}=G(\mathbb{Q})_{+}\cap K$ and let $S(\mathbb{C}):=\Gamma\backslash X^{+}$, which (by the theory of Galois actions on connected components of the canonical model) is defined over a finite abelian extension of $E$ which we will call $E^{S}$. Let $p:X^{+}\rightarrow S(\mathbb{C})$ denote the surjective holomorphic map given by automorphic forms (i.e. $p$ is the quotient map composed with the embedding into projective space). In [18] and [20], $\mathrm{Sh}_{K}(G,X)$ is called a Shimura variety, while the connected component $S$ is called a connected Shimura variety. As our work is only concerned with the connected case, we will use the term “Shimura variety” to refer to connected components, like $S$.

We will indulge in a slight abuse of notation because, as we will almost always be working in $G^{\mathrm{ad}}(\mathbb{Q})^{+}$, we will use the same name for $\Gamma$ and for the image of $\Gamma$ in $G^{\mathrm{ad}}(\mathbb{Q})^{+}$.

Example 4** (The $j$ function).**

Take $G=\mathrm{GL}_{2}$. Let $X$ be the $\mathrm{GL}_{2}(\mathbb{R})$-conjugacy class of the morphism $h:\mathbb{S}\rightarrow\mathrm{GL}_{2,\mathbb{R}}$, which on real points is given by

[TABLE]

Then we can see $X$ as the union of the upper and lower half-planes of $\mathbb{C}$; $X=\mathbb{H}^{+}\cup\mathbb{H}^{-}$. Let $K=\mathrm{GL}_{2}\left(\widehat{\mathbb{Z}}\right)$, so that $\Gamma=\mathrm{SL}_{2}(\mathbb{Z})$. Then the map $p:X^{+}\rightarrow S(\mathbb{C})$ becomes the $j$-function: $j:\mathbb{H}^{+}\rightarrow\mathbb{C}$. Note that this is a case where $\Gamma$ is not torsion-free, but, because it is so well-known, it is a good example to keep in mind as we go along.

Example 5** (Siegel moduli spaces).**

Here we generalise the previous example to construct the moduli space of principally polarised abelian varieties of dimension $g$ (following well-established tradition, for this example we reserve the letter $g$ to be the dimension of the abelian varieties, not an element of the group $G$). Let:

[TABLE]

where $I_{g}$ is the $g\times g$ identity matrix. Consider:

[TABLE]

where $v(M)\in\mathbb{R}^{\times}$. Then $v:\mathrm{GSp}_{2g}\rightarrow\mathbb{G}_{m}$ is a homomorphism of linear algebraic groups. So we can define $\mathrm{Sp}_{2g}$ as the kernel of $v$, that is:

[TABLE]

Just like in the previous example, we define:

[TABLE]

In this case, $X:=\mathrm{GSp}_{2g}(\mathbb{R})\cdot h$ can be identified with the disjoint union $\mathbb{H}_{g}^{+}\cup\mathbb{H}_{g}^{-}$, where:

[TABLE]

is called the Siegel upper half-space. $\mathbb{H}_{g}^{-}$ is defined in the same way, but changing $B>0$ for $B<0$. The action of $\mathrm{GSp}_{2g}(\mathbb{R})$ on $\mathbb{H}_{g}^{+}\cup\mathbb{H}^{-}$ is given by:

[TABLE]

Now we take $K=\mathrm{GSp}_{2g}\left(\widehat{\mathbb{Z}}\right)$, so $\Gamma=\mathrm{Sp}_{2g}(\mathbb{Z})$. The quotient $\mathcal{A}_{g}(\mathbb{C}):=\Gamma\backslash\mathbb{H}_{g}^{+}$ is the (coarse) moduli space of principally polarised abelian varieties of dimension $g$. These spaces are also known as Siegel moduli spaces.

Remark 3.5* (Products of Shimura data).*

Let $\left(G_{1},X_{1}\right)$ and $\left(G_{2},X_{2}\right)$ be Shimura data, with associated connected Shimura varieties $S_{1}(\mathbb{C})=\Gamma_{1}\backslash X_{1}^{+}$ and $S_{2}(\mathbb{C})=\Gamma_{2}\backslash X_{2}^{+}$. Then $\left(G_{1}\times G_{2},X_{1}\times X_{2}\right)$ is a Shimura datum, $X_{1}^{+}\times X_{2}^{+}$ is a connected component of $X_{1}\times X_{2}$ whose stabiliser in $(G_{1}\times G_{2})^{\mathrm{ad}}(\mathbb{R})=(G_{1}^{\mathrm{ad}}\times G_{2}^{\mathrm{ad}})(\mathbb{R})$ is $G_{1}^{\mathrm{ad}}(\mathbb{R})^{+}\times G_{2}^{\mathrm{ad}}(\mathbb{R})^{+}$. And so we can see $S_{1}\times S_{2}$ as the Shimura variety with datum $\left(G_{1}\times G_{2},X_{1}\times X_{2}\right)$ whose complex points are given by $\left(\Gamma_{1}\backslash X_{1}^{+}\right)\times\left(\Gamma_{2}\backslash X_{2}^{+}\right)$.

3.3. Classifying Points

Let $(G,X)$ be a Shimura datum. From the definition, a point $x\in X$ corresponds to a homomorphism $x:\mathbb{S}\rightarrow G_{\mathbb{R}}$. We can then classify the points of $X^{+}$ in terms of the image of the homomorphism they represent in the following way: the Mumford-Tate group of $x$, denoted $\mathrm{MT}(x)$, is the smallest algebraic subgroup $H$ of $G$ defined over $\mathbb{Q}$ such that $x$ factors through $H_{\mathbb{R}}$.

Definition**.**

Let $x\in X^{+}$ and $p(x)=z\in S(\mathbb{C})$. The Hecke-orbit of $z$ is the set of points $z^{\prime}\in S(\mathbb{C})$ for which there exists $g\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$ such that $p(gx)=z$.

Points in the same Hecke-orbit will have the same Mumford-Tate group.

Definition**.**

A point $x\in X$ is said to be special if $\mathrm{MT}(x)$ is a subtorus of $G$. On the other hand, $x$ is called Hodge-generic if $\mathrm{MT}(x)=G$.

If we want to develop model theory of Shimura varieties, we should use a language that is capable of recognising points of $X^{+}$ with different Mumford-Tate group (this is done in Remark 4.4). We will show some results in the following subsection on special subvarieties that will eventually allow us to do that. In the mean time, we recall two results that show that the action of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ on $X^{+}$ can already distinguish between special points and Hodge-generic points in the case of pure Shimura varieties.

Theorem 3.6** (see [10, Theorem 2.3]).**

Let $(G,X)$ be a pure Shimura datum and let $X^{+}$ be a connected component of $X$. For any $x\in X^{+}$ the following are equivalent:

1. (1)

$x$* is a special point.* 2. (2)

There exists $g\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$ with the property that $x$ is the unique fixed point of $g$ in $X^{+}$.

Proposition 3.7** (see [27, Lemma 2.2]).**

Let $(G,X)$ be a pure Shimura datum and let $X^{+}$ be a connected component of $X$. Let $x\in X^{+}$ be Hodge-generic and suppose $g\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$ fixes $x$. Then $g$ is the identity.

Example 6** (Special points of $\mathcal{A}_{g}$).**

Using Theorem 3.6 we know that the points of $\mathbb{H}^{+}_{g}$ which are special are those which are fixed by some element of $\mathrm{GSp}_{2g}(\mathbb{Q})^{+}$. These are precisely the points that correspond to principally polarised abelian varieties with complex multiplication (CM for short).

3.4. Special Subvarieties

Let $(G,X)$ and $(H,Y)$ be Shimura data. Let $\phi:H\rightarrow G$ be a homomorphism of algebraic groups over $\mathbb{Q}$ that induces a map $Y^{+}\rightarrow X^{+}$ given by $y\mapsto\phi\circ y$. In this case we say that $\phi:(H,Y^{+})\rightarrow(G,X^{+})$ is a morphism of connected Shimura data. If furthermore $\phi$ is an inclusion, then we say that $(H,Y^{+})$ is a connected Shimura subdatum of $(G,X^{+})$.

Let $\Gamma^{G}\leq G$ and $\Gamma^{H}\leq H$ be congruence subgroups and let $S_{1}=\Gamma^{H}\backslash Y^{+}$ and $S_{2}=\Gamma^{G}\backslash X^{+}$ be the corresponding connected Shimura varieties. If in addition to being a morphism of Shimura varieties, $\phi\left(\Gamma^{H}\right)\subseteq\Gamma^{G}$, then $\phi$ induces a map in the quotients $[\phi]:\Gamma^{H}\backslash Y^{+}\rightarrow\Gamma^{G}\backslash X^{+}$ given by $[y]\mapsto[\phi\circ y]$ which is well-defined, holomorphic, and algebraic with respect to the algebraic structures of $S_{1}$ and $S_{2}$ (see [20, Fact 2.6] for references). The morphisms $S_{1}\rightarrow S_{2}$ which are obtained in this way are called morphisms of Shimura varieties.

Definition**.**

A special subvariety of a Shimura variety $S$ is an irreducible component of the image of a morphism of Shimura varieties $T\rightarrow S$. A special set in $S$ is a finite union of special subvarieties.

This definition does not conflict with the definition of special point we gave in the previous subsection. The Mumford-Tate group of a point $x\in X^{+}$ is a torus if and only if $p(x)\in S(\mathbb{C})$ is a zero-dimensional special subvariety (see [9, §18], [18, Remark 12.6], or [20, §4]).

Example 7** (Special subvarieties of $\mathbb{C}^{2}$).**

Let us start by looking at $j:\mathbb{H}\rightarrow\mathbb{C}$ as a Shimura variety. Given that is it one-dimensional, $\mathbb{C}$ does not have special subvarieties other than the special points. We can instead consider $j:\mathbb{H}^{2}\rightarrow\mathbb{C}^{2}$, where $j$ is now defined on each component, which gives us a Shimura surface, and so we can look at its one-dimensional special subvarieties. These consist of the subvarieties cut out by the family $\left\{\Phi_{N}(X,Y)\right\}_{N\in\mathbb{N}}$ of modular polynomials of the $j$ function, plus the subvarieties of the form $\left\{\tau\right\}\times\mathbb{C}$ or $\mathbb{C}\times\left\{\tau\right\}$, where $\tau$ is a special point. This result will be generalised in Lemma 3.22.

Remark 3.8*.*

Let $S$ be a Shimura variety and $V\subseteq S$ be a special subvariety. By definition this means that it can be realised as the image of a morphism of Shimura varieties $[\phi]:T\rightarrow S$. If $(G,X^{+})$ is the connected Shimura datum of $S$, let $H$ be the image of $\phi$ in $G$. Therefore $H$ is a $\mathbb{Q}$-subgroup of $G$. Also, let $X^{+}_{V}$ be the image of $\phi$ in $X^{+}$. Thus $\left(H,X^{+}_{V}\right)$ is a connected Shimura datum and, if $\Gamma$ is the congruence subgroup defining $S$, then $\Gamma^{H}:=H(\mathbb{Q})_{+}\cap\Gamma$ is a congruence subgroup of $H$ such that the inclusion $i:H\hookrightarrow G$ defines a morphism between the connected Shimura varieties $\Gamma^{H}\backslash X_{V}^{+}$ and $S$ with image $V$. In other words, we can always find a subgroup of $G$ that will define a Shimura datum for $V$.

We hope that the following commutative diagram becomes a useful visual aid to help the reader with the notation we have introduced so far:

{X^{+}_{V}\subseteq X^{+}}$${\Gamma^{H}\backslash X^{+}_{V}}$${V\subseteq S(\mathbb{C})}$$\scriptstyle{p^{H}}$$\scriptstyle{p}$$\scriptstyle{[i]}

Definition**.**

Let $V\subseteq S$ be a special subvariety. A point $x\in X^{+}$ is said to be Hodge-generic in $V$ if $V$ is the smallest special subvariety of $S$ containing $p(x)$. Given a subset $A\subseteq S(F)$ (where $F$ is some algebraically closed field of characteristic zero), the special closure of $A$, denoted $\mathrm{spcl}(A)$, is the smallest special set of $S(F)$ that contains $A$. So a point is always Hodge-generic in its special closure.

Remark 3.9*.*

The special closure of a point $x\in X^{+}$ is given by its Mumford-Tate group. This is because $\left(\mathrm{MT}(x),\mathrm{MT}(x)(\mathbb{R})\cdot x\right)$ is a Shimura subdatum of $(G,X)$, and so the image of $\left(\mathrm{MT}(x)(\mathbb{R})^{+}\cdot x\right)\times\left\{g\right\}$ in $\mathrm{Sh}_{K}(G,X)(\mathbb{C})$ defines the smallest special subvariety containing $[x,g]_{K}$.

In what remains of this subsection, fix a Shimura variety $p:X^{+}\rightarrow S(\mathbb{C})$, let $\Sigma$ be the set of coordinates of special points of $S(\mathbb{C})$ and let $E^{S}$ be the minimal abelian extension of the reflex field associated to the Shimura datum of $S$ over which $S(\mathbb{C})$ is defined.

Remark 3.10* (see [20, Proposition 4.14]).*

Every special subvariety contains a dense set of special points, so every special subvariety is defined as a variety over $E^{S}(\Sigma)$.

Now we define the notion of special domain. Essentially, a special domain for a special subvariety $V\subseteq S$ is a subset of $X^{+}$ that can act as the connected component of some Shimura datum which defines $V$.

Definition**.**

Let $V\subseteq S^{n}$ be a special subvariety and choose a $\mathbb{Q}$-subgroup $H$ of $G^{n}$ and a point $x\in p^{-1}(V)$. So $(H,H(\mathbb{R})\cdot x)$ is a Shimura subdatum of $(G^{n},X^{n})$ for $V$. This can be done more canonically (but not completely) if we choose $x\in p^{-1}(V)$ to be Hodge-generic in $V$ and then $H$ can be taken to be the Mumford-Tate group of $x$. In this case we call $H(\mathbb{R})\cdot x$ a special domain for $V$.

Note that if we choose a point $y\in p^{-1}(V)$ different from $x$ that is Hodge-generic in $V$, then $H(\mathbb{R})\cdot y$ is another special domain for $V$ which is either equal to $H(\mathbb{R})\cdot x$ or disjoint from it. In fact, for every special domain $X^{+}_{V}\subseteq\left(X^{+}\right)^{m}$ of $V$, there is $\gamma\in\Gamma^{m}$ such that $\gamma X^{+}_{V}=\left(H(\mathbb{R})\cdot x\right)^{+}$. Therefore, we can find at most countably many different special domains for $V$, so we will number them and denote them by $X^{+}_{V,i}$, where $i$ is a natural number between $1$ and the number of special domains for $V$ (usually $\omega$). This way if $i\neq j$, then $X^{+}_{V,i}\cap X^{+}_{V,j}=\emptyset$.

Remark 3.11*.*

Let $X_{V,i}^{+}$ be a special domain, and suppose $\gamma\in\Gamma$ is such that $\gamma X_{V,i}^{+}=X_{V,i}^{+}$. If $H$ is the subgroup of $G$ corresponding to $X_{V,i}^{+}$, then $\gamma\in H(\mathbb{Q})$, because a consequence of the axioms of Shimura datum is that $H(\mathbb{R})^{+}$ maps surjectively to the group of biholomorphisms of $X^{+}_{V,i}$ (see [18, Proposition 4.8]).

Usually, the order in which we index the special domains will not be important, but we will make one distinction. When we define the special subvarieties $Z^{V,i}_{\overline{g}}$ in §3.6, we will want to index the special domains of $Z_{\overline{g}}^{V,i}$ correspondingly with the special domains of $V$, as it will be explained in Remark 3.23. One could define a more intrinsic form of indexing the special domains, but as we will not really use this extra information, we have opted for a simpler form of notation.

Definition**.**

A special set in $\left(X^{+}\right)^{m}$ is a finite union of special domains. The special closure of $B\subseteq\left(X^{+}\right)^{m}$ is the smallest special set containing $B$, which again will be denoted by $\mathrm{spcl}(B)$.

Example 8** (Trivial special closures).**

If $x\in X^{+}$ is Hodge-generic, then $\mathrm{spcl}(x)=X^{+}$ and $\mathrm{spcl}(p(x))=S(\mathbb{C})$. If $x\in X^{+}$ is special, then $\mathrm{MT}(x)$ is a torus, and as it is commutative, we have that $\mathrm{MT}(x)(\mathbb{R})^{+}\cdot x=\left\{x\right\}$. So, for some $i$, $X_{x,i}^{+}=\left\{x\right\}=\mathrm{spcl}(x)$ and $\mathrm{spcl}(p(x))=\left\{p(x)\right\}$.

Remark 3.12* (A Word on Reflex Fields).*

We conclude this subsection with an observation about reflex fields. Let $V\subseteq S$ be a special subvariety and let $X^{+}_{V,i}$ be a special domain for $V$. Let $\Gamma^{H}:=\Gamma\cap H(\mathbb{Q})_{+}$. Then we can view $V$ as a the image of a Shimura variety as: $S_{H}=\Gamma^{H}\backslash X^{+}_{V,i}$ under a Shimura morphism. Let $E^{S_{H}}$ be the (finite abelian extension of the) reflex field of $S_{H}$. By [18, Remark 12.3], $E^{S}\subseteq E^{S_{H}}$. By the general theory of canonical models ([18, §12]), we have that $E^{S_{H}}\subseteq E^{S}(\Sigma)$.

3.5. Galois Representations I

Following [27, §2] and [20, §6], we describe Galois representations attached to points in Shimura varieties, and we will end by stating Pink’s conjecture, which will later influence the way we state conditions FIC 1 & 2.

Given a congruence subgroup $\Gamma^{\prime}$ of $\Gamma$ we have a morphism of connected Shimura varieties $[\mathrm{id}]:\Gamma^{\prime}\backslash X^{+}\rightarrow\Gamma\backslash X^{+}$ induced by the identity map $\mathrm{id}:G\hookrightarrow G$. So as to distinguish elements in the quotients of $X^{+}$ with respect to the different congruence groups, let us denote by $[x]_{\Gamma}$ the class of $x\in X^{+}$ in the quotient $\Gamma\backslash X^{+}$. Remember we are assuming $\Gamma$ is neat.

Lemma 3.13**.**

For every $\gamma\in\Gamma$ and every $x\in X^{+}$, $\gamma x\neq x$.

Proof.

Suppose that $\gamma x=x$ for some $x\in X^{+}$ and some $\gamma\in\Gamma$. By definition $\Gamma=G(\mathbb{Q})_{+}\cap K$, but if $\gamma x=x$, then this also means that $\gamma$ belongs to $Z_{G_{\mathbb{R}}}(x(\mathbb{S}))(\mathbb{R})$. As $\Gamma$ is neat, we can use [27, Lemma 2.2] and [27, Lemma 2.3] to conclude straightaway that $\gamma=1$. ∎

Choose a point $z\in S(\mathbb{C})=\Gamma\backslash X^{+}$ and choose $x\in X^{+}$ so that $[x]_{\Gamma}=z$. Lemma 3.13 shows that $[\mathrm{id}]^{-1}(z)$ carries a transitive $\Gamma$-action. If $\Gamma^{\prime}$ is normal in $\Gamma$, then this action factors through $\Gamma/\Gamma^{\prime}$, and can be described by: for $y\in X^{+}$ such that $[y]_{\Gamma}=[x]_{\Gamma}$ and for every $\gamma\in\Gamma$, define $\gamma[y]_{\Gamma^{\prime}}:=[\gamma y]_{\Gamma^{\prime}}$. The action of $\Gamma/\Gamma^{\prime}$ on $[\mathrm{id}]^{-1}(z)$ is transitive. Thus we obtain:

Corollary 3.14**.**

The action of $\Gamma/\Gamma^{\prime}$ on $[\mathrm{id}]^{-1}(z)$ is simply transitive.

The varieties of the form $\Gamma^{\prime}\backslash X^{+}$, where $\Gamma^{\prime}$ is a normal congruence subgroup of $\Gamma$, form an inverse system with morphisms the maps $[\mathrm{id}]$ described above. Let $\mathbb{X}^{+}$ be the inverse limit of the system. Any $\tilde{x}\in\mathbb{X}^{+}$ can be denoted as $([x_{i}]_{\Gamma_{i}})_{\Gamma_{i}}$, so that if $\Gamma_{i}\subseteq\Gamma_{j}$, then $[x_{i}]_{\Gamma_{j}}=[x_{j}]_{\Gamma_{j}}$. Also, there are natural maps

[TABLE]

compatible with the maps of the system. As we mentioned before, the Shimura variety $\mathrm{Sh}_{K}(G,X)$ is canonically defined over its reflex field $E$, which is independent of the choice of $K$. But when we look at a connected component of this variety like $S(\mathbb{C})=\Gamma\backslash X^{+}$, this variety is defined over a finite abelian extension of $E$ that we have denominated $E^{S}$. The map $[\mathrm{id}]:S^{\prime}(\mathbb{C})=\Gamma^{\prime}\backslash X^{+}\rightarrow S(\mathbb{C})$ is defined over the compositum of the fields $E^{S^{\prime}}$ and $E^{S}$. And so, the whole inverse system is defined over $E^{\mathrm{ab}}$, the maximal abelian extension of $E$.

Let $L\subseteq\mathbb{C}$ be a finitely generated field extension of $E^{\mathrm{ab}}$ such that $z$ is defined over $L$. Let $\overline{L}$ denote the algebraic closure of $L$ in $\mathbb{C}$. Then the Galois group $\mathrm{Gal}\left(\overline{L}/L\right)$ also acts on the fibre over $z$ of the morphism $[\mathrm{id}]:\Gamma^{\prime}\backslash X^{+}\rightarrow\Gamma\backslash X^{+}$. We would like to compare the actions of $\mathrm{Gal}\left(\overline{L}/L\right)$ and $\Gamma/\Gamma^{\prime}$ on $[\mathrm{id}]^{-1}(z)$. For this we need some elementary results.

Lemma 3.15**.**

Let $H$ and $G$ be groups and $X$ be a set with a left $G$-action and a compatible right $H$-action. If the action of $H$ is transitive and free, then, fixing a point $x_{0}\in X$, there is a unique group homomorphism $\rho_{x_{0}}:G\rightarrow H$ such that for every $g\in G$ $gx_{0}=x_{0}\rho_{x_{0}}(g)$. Moreover, if $x_{1}=x_{0}h$, then the associated homomorphism $\rho_{x_{1}}$ satisfies $\rho_{x_{1}}=h^{-1}\rho_{x_{0}}h$.

Proof.

As $H$ acts transitively on $X$, for every $g\in G$ there is $h\in H$ such that $gx_{0}=x_{0}h$. If $h_{1},h_{2}\in H$ are such that $x_{0}h_{1}=x_{0}h_{2}$, then $x_{0}$ is a fixed point of $h_{1}h_{2}^{-1}$, but as the action of $H$ is free, then $h_{1}=h_{2}$. Therefore we can define a map $\rho_{x_{0}}:G\rightarrow H$ by setting $\rho_{x_{0}}(g)=h$, where $gx_{0}=x_{0}h$.

If $g_{1}x_{0}=x_{0}h_{1}$ and $g_{2}x_{0}=x_{0}h_{2}$, then $g_{1}g_{2}x_{0}=g_{1}x_{0}h_{2}=x_{0}h_{1}h_{2}$, and so $\rho_{x_{0}}$ is a group homomorphism.

Finally, if $x_{1}=x_{0}h$, then $x_{1}h^{-1}\rho_{x_{0}}(g)h=x_{0}\rho_{x_{0}}(g)h=gx_{0}h=gx_{1}$, and so $\rho_{x_{1}}=h^{-1}\rho_{x_{0}}h$. ∎

Although $\Gamma$ acts on the left on $X^{+}$, we can define a corresponding right action of $\Gamma$ on $X^{+}$ in the usual way by $x\gamma:=\gamma^{-1}x$, for all $x\in X^{+}$ and $\gamma\in\Gamma$.

Lemma 3.16**.**

Assuming that $\Gamma^{\prime}$ is a normal congruence subgroup of $\Gamma$, the actions of $\mathrm{Gal}\left(\overline{L}/L\right)$ and $\Gamma/\Gamma^{\prime}$ on $[\mathrm{id}]^{-1}(z)$ commute.

Proof.

Every $\gamma\in\Gamma$ defines a homomorphism $G\rightarrow G$ given by $g\mapsto\gamma g\gamma^{-1}$, which itself induces a morphism of Shimura data $(G,X)\rightarrow(G,X)$ which on $X$ is given by $x\mapsto\gamma x$. As $\Gamma^{\prime}$ is normal in $\Gamma$, this in turn induces a morphism of Shimura varieties $[\gamma]:S^{\prime}(\mathbb{C})=\Gamma^{\prime}\backslash X^{+}\rightarrow S(\mathbb{C})=\Gamma^{\prime}\backslash X^{+}$. This morphism is defined over the compositum $E^{S}E^{S^{\prime}}$. As $L$ contains $E^{S}E^{S^{\prime}}$, then $[\gamma]$ is fixed under the action of any element of $\mathrm{Gal}\left(\overline{L}/L\right)$, which proves the statement. ∎

With these results we can now compare the actions of $\mathrm{Gal}\left(\overline{L}/L\right)$ and $\Gamma/\Gamma^{\prime}$ on $[\mathrm{id}]^{-1}(z)$. Choosing $[y]_{\Gamma^{\prime}}\in[\mathrm{id}]^{-1}(z)$, by Lemma 3.15 (whose conditions are ensured by Lemma 3.16) we obtain a homomorphism

[TABLE]

We remark that for this we only need $L$ to contain $E^{S}$, $E^{S^{\prime}}$ and the coordinates of $z$; it is not necessary that $L$ contain all of $E^{\mathrm{ab}}$.

Let $\hat{\Gamma}=\varprojlim\Gamma/\Gamma^{\prime}$, where the limit is taken over the normal congruence subgroups $\Gamma^{\prime}$ of $\Gamma$. Then both $\mathrm{Gal}\left(\overline{L}/L\right)$ and $\hat{\Gamma}$ act on the fibre $\pi_{\Gamma}^{-1}(z)$. In fact the action of $\hat{\Gamma}$ can be understood more generally in the following way. For the Shimura datum $(G,X)$ define the space

[TABLE]

whose elements we denote as $[x,g]$, and for $K$ a compact open subgroup of $G(\mathbb{A}_{f})$, let

[TABLE]

When $K$ is neat, the fibre over any point of $\mathrm{Sh}_{K}(G,X)$ under $\pi_{K}$ has a simply transitive right action by $K$ (see [27, Lemma 2.1]) given by $[x,g]k:=[x,gk]$. On the other hand, there is an injective map $X^{+}\rightarrow\mathrm{Sh}(G,X)$ given by $x\mapsto[x,1]$. Observe that for all $\gamma\in\Gamma$ we have that $[\gamma x,1]=[x,\gamma^{-1}]$. The action of $\hat{\Gamma}$ on $\pi_{\Gamma}^{-1}(z)$ is then just the restriction of the action of $K$, and so $\hat{\Gamma}$ acts simply transitively on $\pi_{\Gamma}^{-1}(z)$. Choosing a point $\tilde{y}\in\pi_{\Gamma}^{-1}(z)$, we get a continuous homomorphism:

[TABLE]

Remark 3.17*.*

In the case that $z$ is a special point, the homomorphism $\rho_{\tilde{y}}$ can be described using the reciprocity map coming from the theory of complex multiplication (see [27, Remark 2.7]).

Definition**.**

With the above notation, $z\in S(\mathbb{C})$ is called Galois generic if the image of $\rho_{\tilde{y}}$ is open in $\hat{\Gamma}$ (observe that openness of the image does not depend on the choice of $\tilde{y}$). The point $z$ is called strictly Galois generic if the image of $\rho_{\tilde{y}}$ equals $\hat{\Gamma}$.

The following proposition says that the notion of “Galois generic” is well-defined.

Proposition 3.18** (see [20, Proposition 6.4]).**

The notion of being ‘Galois generic’ is independent of the choice of $L$.

Proposition 3.19** (see [20, Proposition 6.7]).**

Every Galois generic point on $S(\mathbb{C})$ is Hodge generic.

We are now in a position to state Pink’s conjecture, which is the converse of the last proposition.

Conjecture 1** (see [20, Conjecture 6.8]).**

Every Hodge generic point on $S(\mathbb{C})$ is Galois generic.

In the case of Siegel moduli spaces, [20, Remark 6.10] and [27, p. 265] explain the relation between this conjecture and the Mumford-Tate conjecture, concluding that adèlic refinements of Mumford-Tate imply Conjecture 1. A result of Serre provides such a refinement for some cases:

Theorem 3.20** (see [20, Theorem $6.13$]).**

Let $(\mathrm{GSp_{2g},\mathbb{H}_{2g}})$ be the connected Shimura data of Example 5. Assume that $g$ is odd or equal to $2$ or $6$. Then for any suitable congruence subgroup $\Gamma$, the Hodge-generic points of the corresponding Shimura variety $S(\mathbb{C})$ are Galois generic.

3.6. Galois Representations II

Here we will describe the aspects of Galois representations described in §3.5 that can be “seen” by the model theory. Essentially, the difference with the previous treatment is that we need to restrict the normal subgroups $\Gamma^{\prime}$ of $\Gamma$ to be of a specific form. The results of §3.7 show that this difference is merely technical, and does not affect the results nor the general theory of Galois representations attached to points.

Let $S$ be a Shimura variety with Shimura datum $\left(G,X^{+}\right)$ and $S(\mathbb{C})=\Gamma\backslash X^{+}$.

Proposition 3.21** (see [21, §4.1 Corollary 1]).**

Let $\Gamma$ be an arithmetic subgroup of an algebraic group $G$ defined over $\mathbb{Q}$. Then for any $g\in G(\mathbb{Q})$, $g\Gamma g^{-1}$ is also an arithmetic subgroup of $G(\mathbb{Z})$.

In particular $\Gamma\cap g\Gamma g^{-1}$ has finite index in $\Gamma$. Not only that, but choosing $N$ large enough so that the denominators that appear in $g$ and $g^{-1}$ are coprime with $N$, we get that $\Gamma(N)\subseteq\Gamma\cap g\Gamma g^{-1}$, and therefore $\Gamma\cap g\Gamma g^{-1}$ is a congruence subgroup of $G(\mathbb{Z})$. In the following Lemma we define the notation for the special subvariety $Z^{V,i}_{\overline{g}}$. These subvarieties play a major role in the model theory of Shimura varieties.

Lemma 3.22** (cf. [10, Lemma 2.6]).**

Let $\overline{g}=(g_{1},\ldots,g_{n})$ be a tuple of elements belonging to $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ and let $X_{V,i}^{+}$ be a special domain in $\left(X^{+}\right)^{m}$. The image of the map

[TABLE]

is a special subvariety which we will denote $Z^{V,i}_{\overline{g}}$.

Proof.

The Shimura datum of $S(\mathbb{C})^{m+n}$ is $\left(G^{m+n},\left(X^{+}\right)^{m+n}\right)$. Let $H$ be the $\mathbb{Q}$-subgroup of $G^{m}$ acting on $X^{+}_{V,i}$ given by the definition of $X^{+}_{V,i}$. Consider the morphism of algebraic groups

[TABLE]

This induces a morphism of connected Shimura data $\phi:(H,X_{V,i}^{+})\rightarrow\left(G^{m+n},\left(X^{+}\right)^{m+n}\right)$. Let $K$ be a compact open subgroup of $G(\mathbb{A}_{f})$ such that if $\Gamma=G(\mathbb{Q})_{+}\cap K$, then $S(\mathbb{C})=\Gamma\backslash X^{+}$. Then $S(\mathbb{C})^{m+n}=\Gamma^{m+n}\backslash\left(X^{+}\right)^{m+n}$.

Let $\Gamma^{\prime}=\left(g_{1}^{-1}\Gamma g_{1}\cap\cdots\cap g_{n}^{-1}\Gamma g_{n}\right)^{m}\cap H(\mathbb{A}_{f})$. Notice that $\phi(\Gamma^{\prime})\subseteq\Gamma^{m+n}$. Therefore, $\phi$ induces a morphism of connected Shimura varieties $\Gamma^{\prime}\backslash X^{+}_{V,i}\rightarrow\Gamma^{m+n}\backslash\left(X^{+}\right)^{m+n}$ whose image is precisely:

[TABLE]

By definition, $Z_{\overline{g}}^{V,i}$ is a special subvariety. ∎

Remark 3.23* (Indexing of the special domains of $Z_{\overline{g}}^{V,i}$).*

Given the way we have constructed the special domains of $Z_{\overline{g}}^{V,i}$ in Lemma 3.22, we will fix the indexing and say that $X^{+}_{Z_{\overline{g}}^{V,i},i}$ is the special domain of $Z^{V,i}_{\overline{g}}$ which is parametrised by $X^{+}_{V,i}$. This means that for any $\overline{x}\in X^{+}_{Z_{\overline{g}}^{V,i},i}$ there is $y\in X^{+}_{V,i}$ such that $\overline{x}=\overline{g}y$, so we can write $X^{+}_{Z_{\overline{g}}^{V,i},i}=\overline{g}X^{+}_{V,i}$. Later, when we define Shimura structures in §4, it will be evident that this parametrisation is definable in the language of Shimura structures.

Remark 3.24*.*

When $X_{V,i}^{+}=X^{+}$ (and so $V=S$), we will simplify the notation to be $Z_{\overline{g}}:=Z_{\overline{g}}^{S}$..

The special subvarieties $Z_{\overline{g}}^{V,i}$ encode a lot of the information on Galois representations attached to points in $S$, and they will become essential in our model theoretic approach. It is rather crucial to understand how they relate to each other, so in what remains of this section we will recall the treatment done in [10, §2.5] and extend it to the subvarieties $Z_{\overline{g}}^{V,i}$.

Let $\overline{g}=\left(e,g_{1},\ldots,g_{n}\right)$ be a tuple of distinct elements of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$. The corresponding variety $Z_{\overline{g}}$ is biholomorphic to $\Gamma_{\overline{g}}\backslash X^{+}$, where:

[TABLE]

Suppose $\left\{h_{1},\ldots,h_{m}\right\}$ is a subset of $G$ such that $\left\{g_{1},\ldots,g_{n}\right\}\subseteq\left\{h_{1},\ldots,h_{m}\right\}$. Let $\overline{h}:=(e,h_{1},\ldots,h_{m})$. Then there is a finite morphism

[TABLE]

induced by the natural map $\Gamma_{\overline{h}}\backslash X^{+}\rightarrow\Gamma_{\overline{g}}\backslash X^{+}$. If $\Gamma_{\overline{h}}$ is normal in $\Gamma_{\overline{g}}$, then the fibres carry a simply transitive action of $\Gamma_{\overline{g}}/\Gamma_{\overline{h}}$. For the remainder of this section, we will only consider the case when $\Gamma_{\overline{h}}$ is normal in $\Gamma_{\overline{g}}$.

The action of $\Gamma_{\overline{g}}/\Gamma_{\overline{h}}$ on the fibers of $\psi_{\overline{h},\overline{g}}$ in $Z_{\overline{h}}$ is given by regular maps defined over $E^{S}(\Sigma)$. Choose $z\in Z_{\overline{g}}$ and let $L$ be a finitely generated field extension of $E^{S}(\Sigma)$ such that $z$ is defined on $L$. Then $\mathrm{Aut}(\mathbb{C}/L)$ acts on the fibre of $\psi_{\overline{h},\overline{g}}$ over $z$ and, because the action of $\Gamma_{\overline{g}}/\Gamma_{\overline{h}}$ is given by regular maps defined over $E^{S}(\Sigma)$, then this action commutes with the action of $\Gamma_{\overline{g}}/\Gamma_{\overline{h}}$. The action of $\Gamma_{\overline{g}}/\Gamma_{\overline{h}}$ on the fibres of $\psi_{\overline{h},\overline{g}}$ is transitive and (as $\Gamma$ is torsion free in $G^{\mathrm{ad}}(\mathbb{Q})^{+}$) free. Therefore, just as in §3.5, for any point in the fibre above $z\in Z_{\overline{g}}$ we obtain a continuous homomorphism

[TABLE]

If $\overline{z}=(z_{1},\ldots,z_{m})\in Z_{\overline{g}}^{V,i}\subseteq\left(Z_{\overline{g}}\right)^{m}$ for some special subvariety $V\subseteq\left(S(\mathbb{C})\right)^{m}$, then the map $\psi_{\overline{h},\overline{g}}$ induces a map

[TABLE]

The stabilizer of the fiber of $\psi_{\overline{h},\overline{g}}^{V,i}$ over $\overline{z}$ is a subgroup of $\left(\Gamma_{\overline{g}}/\Gamma_{\overline{h}}\right)^{m}$ which we will denote $\Gamma_{\overline{g},\overline{h}}^{V,i}$. Notice that in this case, if $L$ also contains the coordinates of special points in $S(\mathbb{C})$, the homomorphism $\mathrm{Aut}(\mathbb{C}/L)\rightarrow\left(\Gamma_{\overline{g}}/\Gamma_{\overline{h}}\right)^{m}$ factors through $\mathrm{Aut}(\mathbb{C}/L)\rightarrow\Gamma_{\overline{g},\overline{h}}^{V,i}$.

For every special subvariety $V\subseteq S^{m}$, we have obtained an inverse system of varieties $Z_{\overline{g}}^{V,i}$, where the tuples $\overline{g}$ are taken so that $\Gamma_{\overline{g}}\lhd\Gamma$. This induces an inverse system of groups $\Gamma_{\overline{g},\overline{h}}^{V,i}$. We can therefore define the inverse limits

[TABLE]

where the limits are taken over all tuples of distinct elements $\overline{g}=(e,g_{1},\ldots,g_{n})$ such that $\Gamma_{\overline{g}}$ is normal in $\Gamma$. If $z$ has coordinates in an extension $L$ of $E^{S}(\Sigma)$, then, by our discussion in §3.5, we get the conjugacy class of a continuous homomorphism $\mathrm{Aut}(\mathbb{C}/L)\rightarrow\overline{\Gamma^{V,i}}$. Analogously, if we take a tuple $\overline{z}=(z_{1},\ldots,z_{m})\in S(\mathbb{C})^{n}$ with coordinates in an extension $L$ of $E^{S}(\Sigma)$, then we get the conjugacy class of a continuous homomorphism $\mathrm{Aut}(\mathbb{C}/L)\rightarrow\overline{\Gamma}^{m}$.

3.7. Normal Subgroups in $\Gamma$

We now address the question of whether there are normal subgroups of $\Gamma$ of the form $\Gamma_{\overline{g}}$.

Theorem 3.25** (see [21, §4.1 Theorem 4.2]).**

Let $\Gamma$ be an arithmetic subgroup of an algebraic group $G$ defined over $\mathbb{Q}$. Then $\Gamma$ is finitely presented as an abstract group.

In particular, $\Gamma$ is finitely generated. Therefore, for any $n\in\mathbb{N}$, there are only finitely many subgroups of $\Gamma$ of index $n$ (see e.g. [11, Proposition 5.11]). Let $g\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$. Then $\Gamma^{\prime}:=\Gamma\cap g\Gamma g^{-1}$ has finite index in $\Gamma$ by Proposition 3.21. For every $\gamma\in\Gamma$, $\gamma\Gamma^{\prime}\gamma^{-1}$ has the same index as $\Gamma^{\prime}$ in $\Gamma$. Given that there are only finitely many subgroups of $\Gamma$ of this index, the intersection

[TABLE]

is a finite intersection. On the other hand, it is clear that $\Gamma^{\prime\prime}$ is normal in $\Gamma$. Also $\gamma\Gamma^{\prime}\gamma^{-1}=\Gamma\cap\gamma g\Gamma g^{-1}\gamma^{-1}$. And so $\Gamma^{\prime\prime}=\Gamma_{\overline{g}}$ for some tuple $\overline{g}$ from $G^{\mathrm{ad}}(\mathbb{Q})^{+}$.

Observe that this argument can be replicated for any subgroup of $\Gamma$ of the form $\Gamma_{\overline{g}}$ to obtain a normal subgroup of $\Gamma$ of the form $\Gamma_{\overline{h}}$ such that $\overline{h}$ contains the tuple $\overline{g}$.

4. Shimura Structures

Now we will present our first model-theoretic treatment of Shimura varieties (the second approach will be given in §6). For every Shimura variety $S$ we will define a corresponding Shimura structure. Our treatment differs from [10] in that we have enhanced the language to include names for the special domains of special subvarieties.

Definition**.**

A Shimura structure for $S$ is a two-sorted structure $\mathbf{q}:=\left<\mathbf{D},\mathbf{S},q\right>$ where:

1. (1)

$\mathbf{D}:=\left<D;\left\{g\right\}_{g\in G^{\mathrm{ad}}(\mathbb{Q})^{+}},\mathscr{R}_{D}\right>$ is a $G^{\mathrm{ad}}(\mathbb{Q})^{+}$-covering structure. This means that $D$ is a set with an action of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ (so the elements of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ denote function symbols), and $\mathscr{R}_{D}$ is a countable set of subsets of powers of $D$. The elements of $\mathscr{R}_{D}$ are named $D_{V,i}$, where the indexing comes from the indexing of special domains of $X^{+}$. 2. (2)

$\mathbf{S}:=\left<S(F);\mathscr{R}_{S}\right>$ is a variety structure. This means that $S(F)$ is the set of $F$-points of $S$, where $F$ is an algebraically closed field. Variety structures will be interpreted in the field sort $\mathbf{F}:=\left<F;+,\cdot,F_{0}\right>$, where $F_{0}$ is a countable subfield of $F$, which is algebraic over $\mathbb{Q}$, contains $E^{S}(\Sigma)$, and which is interpreted as a set of constants in $\mathbf{F}$. $\mathscr{R}_{S}$ is the set of all Zariski closed subsets of $S(F)^{n}$ defined over $F_{0}$ for all $n\in\mathbb{N}$. 3. (3)

$q:D\rightarrow S(F)$ is a function.

Every Shimura variety $S$ determines (up to choice of a field of constants $F_{0}$) a natural Shimura structure which we will denote $\mathbf{p}$, where we take $F=\mathbb{C}$, $D=X^{+}$, $q=p$, and $D_{V,i}=X^{+}_{V,i}$. For $F_{0}$ we will normally use $E^{S}(\Sigma)$.

We have implicitly defined the language we will use for Shimura structures (which depends on $S$). Let $\mathcal{L}_{D}$ be the language we have used in the $G^{\mathrm{ad}}(\mathbb{Q})^{+}$-covering sort, $\mathcal{L}_{F}$ the language of the field sort, and let $\mathcal{L}$ be the language of Shimura structures. $F_{0}$ is countable, so $\mathcal{L}$ is countable. Finally, let $\mathrm{Th}(\mathbf{p})$ be the complete first-order theory of the two-sorted structure $\mathbf{p}$ in the language $\mathcal{L}$.

4.1. Theory of the Variety Sort

Let $T:=\mathrm{Th}(S(\mathbb{C}))$, which is the complete first-order theory of $S(\mathbb{C})$ in the language $\mathcal{L}_{F}$. We make a few observations, which are standard results from model theory.

1. (1)

The models of $T$ are bi-interpretable with $F$. So any model of $T$ is of the form $S(F)$, for $F$ an algebraically closed field. So $T$ has quantifier elimination, and hence it has finite Morley rank. 2. (2)

By [24, Theorem 6.2.7], $T$ is totally transcendental. 3. (3)

By [24, Theorem 5.3.3], $T$ has a prime model. In fact, it is $S\left(\overline{\mathbb{Q}}\right)$. 4. (4)

By [24, Theorem 5.2.6], as $T$ is countable and totally transcendental, it is also $\omega$-stable.

Remark 4.1* (Projections of special sets).*

Let $V\subseteq S^{n}$ be a special subvariety. The Shimura datum of $S^{n}$ is $(G^{n},X^{n})$ where $G^{n}$ acts coordinate-wise on $X^{n}$. A coordinate projection $\mathrm{pr}:S(F)^{n}\rightarrow S(F)^{m}$ (where $m<n$) is a Shimura morphism as it is induced by the corresponding coordinate projections $G(\mathbb{R})^{n}\rightarrow G(\mathbb{R})^{m}$ and $X^{n}\rightarrow X^{m}$. Therefore, $\mathrm{pr}(V)$ is a special set.

We can obtain an analogous result for the projections of special domains. Suppose that $\mathrm{pr}(V)=W_{1}\cup\cdots\cup W_{r}$, with each $W_{t}$ a special subvariety. If we take a point in the corresponding coordinate projection $\mathrm{pr}\left(X^{+}_{V,i}\right)$, then this point must belong a special domain of some of the $W_{t}$. So we can write $\mathrm{pr}\left(X^{+}_{V,i}\right)\subseteq X^{+}_{W_{1},j_{1}}\cup\cdots\cup X^{+}_{W_{n},j_{r}}$. This union will consist of exactly one special domain for each $W_{t}$ as $X^{+}_{V,i}$ is connected, and so its projection needs to be connected (remember that distinct special domains of the same special subvariety are disjoint). On the other hand, every point in $W_{t}$ can be lifted to a point in $V$, so we get $\mathrm{pr}\left(X^{+}_{V,i}\right)\subseteq X^{+}_{W_{1},j_{1}}\cup\cdots\cup X^{+}_{W_{n},j_{r}}$.

Remark 4.2*.*

In our model theoretic setting of Shimura structures, our approach has been purely algebraic, effectively forgetting about the analytic structure on both sorts, only retaining the algebraic vareity structure on $S$, and the special structure and the $G^{\mathrm{ad}}(\mathbb{Q})^{+}$-action on $X^{+}$. But one can instead study Shimura varieties again in a two-sorted way from the perspective of o-minimality (so both $X^{+}$ and $S$ are now defined over real closed fields), in order to see part of the analytic structure. This has been done by B. Zilber in [30].

In this paper, Zilber studies the structure of special subvarieties in a very general class of arithmetic varieties using o-minimality. Most results are framed in terms of “weakly special subvarieties” and “weakly special domains”, but one can obtain results about special subvarieties using the fact that a weakly special subvariety is special if and only if it contains a special point. The paper proves that weakly special subvarietes define a Noetherian Zariski structure (see [30, Theorem 4.11]) and it also gives a characterisation of special subvarieties (see [30, Example 6.4]). The paper also proves that the special structure of a Shimura curve has trivial geometry in the model-theoretic sense. This result can be interpreted as saying that if we can only see the special subvarieties on the powers of a Shimura curve and we are given a subset $A$ of the curve, then the only points that are algebraic (in the model-theoretic sense) over $A$ are those that belong to the Hecke-orbit of some point in $A$.

It seems from the end of [30, §3.2] that the author is not sure if mixed Shimura varieties are in the class of arithmetic varieties he is considering. But [13, §10.1] shows that mixed Shimura varieties do satisfy the axioms of [30, §3.1].

4.2. Axiomatisation

We will define the special subvarieties condition (SS), which is an axiom scheme in $\mathrm{Th}(\mathbf{p})$ that speaks about the behaviour of $q$. In the following subsection we will show that SS, along with the full theories of both the domain and the variety sort, axiomatise $\mathrm{Th}(\mathbf{p})$.

First, let us set some notation. If $\overline{g}=(g_{1},\ldots,g_{n})$ is a tuple from $G^{\mathrm{ad}}(\mathbb{Q})^{+}$, then $\overline{g}x=(g_{1}x,\ldots,g_{n}x)$. Whenever we are dealing with powers of the Shimura structure, the covering map will also be denoted by $q$, so that $q(\overline{g}x)=(q(g_{1}x),\ldots,q(g_{n}x))$.

Definition**.**

Given a special subvariety $V$ of $S^{n}$, define:

1. (1)

$\mathrm{SS}^{1}_{V,i}:=\forall\overline{x}\in D_{V,i}\left(q(\overline{x})\in V\right)$. 2. (2)

$\mathrm{SS}^{2}_{V,i}:=\forall z\in V\exists\overline{x}\in D_{V,i}\left(q(\overline{x})=z\right)$.

We define the special subvarieties condition to be the axiom scheme $\mathrm{SS}:=\bigcup_{V,i}\mathrm{SS}^{1}_{V,i}\wedge\mathrm{SS}^{2}_{V,i}$.

Definition**.**

Given a tuple $\overline{g}=(g_{1},\ldots,g_{n})$ of elements in $G^{\mathrm{ad}}(\mathbb{Q})^{+}$, consider the sentences:

1. (1)

$\mathrm{MOD}_{\overline{g},V,i}^{1}:=\forall x\in D_{V,i}\left((q(g_{1}x),\ldots,q(g_{n}x))\in Z_{\overline{g}}^{V,i}\right)$. 2. (2)

$\mathrm{MOD}_{\overline{g},V,i}^{2}:=\forall z\in Z_{\overline{g}}^{V,i}\exists x\in D_{V,i}\left((q(g_{1}x),\ldots,q(g_{n}x))=z\right)$.

Define the modularity condition as the axiom scheme $\mathrm{MOD}:=\bigcup_{\overline{g},V,i}\mathrm{MOD}_{\overline{g},V,i}^{1}\wedge\mathrm{MOD}_{\overline{g},V,i}^{2}.$

Remark 4.3*.*

Observe that MOD follows from SS. To see this recall Remark 3.23, and observe that whenever we have a definable set $A\subseteq D$, we have a corresponding definable set $Z_{\overline{g}}^{q(A)}$ (for any tuple $\overline{g}$ of elements of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$) whose elements all have the form $q(\overline{g}a)$ for some $a\in A$, and there is a definable subset $B$ of $D^{n}$ (where $n$ is the length of $\overline{g}$) such that $q(B)=Z_{\overline{g}}^{q(A)}$, with a definable parametrisation from $A$, i.e. the map $A\rightarrow B$ given by $a\mapsto\overline{g}a$ is definable. This is because we can write in a first-order way the parametrisation of $D_{Z_{\overline{g}}^{V,i},i}$ in terms of $D_{V,i}$: $\overline{y}\in D_{Z_{\overline{g}}^{V,i},i}\iff\exists x\in D_{V,i}(\overline{y}=\overline{g}x)$.

Set $\mathrm{T}(\mathbf{p}):=\mathrm{Th}\left<X^{+};\left\{g\right\}_{g\in G^{\mathrm{ad}}(\mathbb{Q})^{+}},\mathscr{R}_{D}\right>\cup T\cup\mathrm{SS}$. We will show in the next subsection that $\mathrm{T}(\mathbf{p})$ axiomatises $\mathrm{Th}(\mathbf{p})$.

Remark 4.4*.*

For $x\in X^{+}$, note that $\mathrm{qftp}(x)$ “knows” about the Mumford-Tate group of $x$. Recall that the Mumford-Tate group determines the smallest special subvariety of $S(\mathbb{C})$ which contains $p(x)$ (see Remark 3.9). So, if $x_{1},x_{2}\in X^{+}$ have different Mumford-Tate groups, then $\mathrm{qftp}(x_{1})\neq\mathrm{qftp}(x_{2})$.

Remark 4.5*.*

Suppose that $S$ is a pure Shimura variety. As special points of $S$ are zero-dimensional special subvarieties, then SS implies that if $x\in X^{+}$ is a special point and $g_{x}\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$ fixes $x$ and only $x$ (which exists by Theorem 3.6), then

[TABLE]

where we view $p(x)$ as a constant symbol (because the coordinates of $p(x)$ are in $F_{0}$). In [10] this is called the special points condition.

4.3. Quantifier Elimination

Let $p:X^{+}\rightarrow S(\mathbb{C})$ be a Shimura variety and $\mathbf{p}$ its corresponding two-sorted structure. In this section we will show that $\mathrm{Th}(\mathbf{p})$ has quantifier elimination. To do that, we will show that $\mathrm{T}(\mathbf{p})$ has quantifier elimination and is complete. The strategy for this is to use Propositions 2.2 and 2.3.

Proposition 4.6**.**

$\mathrm{T}(\mathbf{p})$* has quantifier elimination and is complete. In particular, $\mathrm{T}(\mathbf{p})=\mathrm{Th}(\mathbf{p})$.*

Proof.

Let $\mathbf{q}:=\left<\mathbf{D},\mathbf{S},q\right>$ and $\mathbf{q}^{\prime}:=\left<\mathbf{D}^{\prime},\mathbf{S}^{\prime},q^{\prime}\right>$ be $\omega$-saturated models of $\mathrm{T}(\mathbf{p})$, and suppose we have $x_{1},\ldots,x_{m}\in D$ and $x_{1}^{\prime},\ldots,x_{m}^{\prime}\in D^{\prime}$ such that $\mathrm{qftp}(x_{1},\ldots,x_{m})=\mathrm{qftp}(x_{1}^{\prime},\ldots,x_{m}^{\prime})$. By Propositions 2.2 and 2.3, we need to show that given $y\in D$, there is $y^{\prime}\in D^{\prime}$ such that $\mathrm{qftp}(x_{1},\ldots,x_{m},y)=\mathrm{qftp}(x_{1}^{\prime},\ldots,x_{m}^{\prime},y^{\prime})$. Let $L$ be the field generated over $E^{S}(\Sigma)$ by the coordinates of the $q(x_{i})$.

Let $D_{V,i}$ be the special closure of $(x_{1},\ldots,x_{m},y)$. Using quantifier elimination of $T$ and $\omega$-saturation of $\mathbf{q}^{\prime}$, we will first show that we can find $y^{\prime}\in D^{\prime}$ so that $(x_{1}^{\prime},\ldots,x_{m}^{\prime},y^{\prime})\in D_{U,i}^{\prime}$ and

[TABLE]

where the unions are taken over all finite tuples $\overline{g}$ of elements of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$. To see the equality of (4.1) note that for a tuple $\overline{g}$, $\mathrm{qftp}\left(q(\overline{g}x_{1}),\ldots,q(\overline{g}x_{m}),q(\overline{g}y)/L\right)$ is determined by the smallest algebraic variety $W$ defined over $L$ which contains the point $\left(q(\overline{g}x_{1}),\ldots,q(\overline{g}x_{m}),q(\overline{g}y)\right)$. Notice that $W\subseteq Z_{\overline{g}}^{V,i}$.

Let $\mathrm{pr}$ be the coordinate projection that sends $(\overline{g}x_{1},\ldots,\overline{g}x_{m},\overline{g}y)\mapsto(\overline{g}x_{1},\ldots,\overline{g}x_{m})$. Applying $\mathrm{pr}$ to $\overline{g}D_{V,i}$ (remember Remark 4.1) will give us finitely many special domains, say $D_{U_{1},j_{1}},\ldots,D_{U_{r},j_{r}}$ (which can be proven in $T(\mathbf{p})$). On the other hand, by quantifier elimination of $T$, $\mathrm{pr}(W)$ is formed of finitely many subvarieties defined over $L$, say $W_{1},\ldots,W_{s}$. Therefore, the formula

[TABLE]

is equivalent to the quantifier-free formula

[TABLE]

As this formula is quantifier-free and it is satisfied by $(x_{1},\ldots,x_{m})$, it is also satisfied by $(x_{1}^{\prime},\ldots,x_{m}^{\prime})$, and therefore there is $y^{\prime}\in D^{\prime}$ such that $(x_{1}^{\prime},\ldots,x_{m}^{\prime},y^{\prime})\in D_{V,i}^{\prime}$ and

[TABLE]

Not only that, but as $\overline{g}D_{V,i}=\mathrm{spcl}(\overline{g}x_{1},\ldots,\overline{g}x_{m},\overline{g}y)$, then we have that $W\subseteq Z_{\overline{g}}^{V,i}$, and so $(\overline{g}x^{\prime}_{1},\ldots,\overline{g}x^{\prime}_{m},\overline{g}y^{\prime})\in D^{\prime}_{V,i}$. This means that the left-hand-side of (4.1) plus the set of formulas

[TABLE]

form a finitely satisfiable set in $\mathbf{q}^{\prime}$, so $\omega$-saturation gives us the $y^{\prime}$ we wanted.

Now we show that $\omega$-saturation allows us to choose $y^{\prime}$ so that, in addition, $D_{V,j}^{\prime}$ is the special closure of $\left(\overline{g}x_{1}^{\prime},\ldots,\overline{g}x_{m}^{\prime},\overline{g}y^{\prime}\right)$. A special domain cannot be covered by finitely many proper special subdomains, so we can avoid finitely many proper special subdomains of $D_{V,i}^{\prime}$ when choosing $y^{\prime}$ above. This finishes the proof. ∎

4.4. Special Locus

Definition**.**

For $D_{V,i}\subseteq D^{m}$ and $a_{1},\ldots,a_{n}\in D$, where $n<m$, let $D_{V,i}(\overline{a})$ denote the set of tuples $\overline{y}=(y_{1},\ldots,y_{m-n})$ such that $(\overline{y},\overline{a})\in D_{V,i}$. Let $y_{1},\ldots,y_{m}\in D$ and $A\subseteq D$. We define the special locus of the tuple $\overline{y}$ over $A$, denoted $\mathrm{sploc}(\overline{y}/A)$, to be the smallest set of the form $D_{V,i}(\overline{a})$ containing $\overline{y}$, where $\overline{a}$ is ranges through all finite tuples of $A$.

The following Lemma shows that the special locus is well-defined.

Lemma 4.7**.**

The special locus always exists and, if $C\subseteq D^{m}$ is the special locus of $\overline{y}$ over $A$, then $C$ is definable over $A$ and $q(C)$ is a subvariety of $S^{m}$ defined over $E^{S}(\Sigma,q(A))$.

Proof.

We will assume $m=1$; the same proof goes through for higher dimensions. We will preserve the notation from the definition of special locus. To see that the special locus exists, consider first a set of the form $D_{V,i}(\overline{a})$. We know that $V=q(D_{V,i})$ is a special subvariety of $S^{n+1}$. Intersection with the hyperplanes $X_{k}=q(a_{k})$, $k=2,\ldots,n+1$, is still a subvariety (although probably not special), and then if we project onto the first coordinate we get that the set $q(D_{V,i}(\overline{a}))$ is a (possibly reducible) subvariety of $S$. So an arbitrary intersection of sets of the form $D_{V,i}(\overline{a})$ is actually a finite intersection. The finite intersection of special domains can be written as a finite union of some other special domains. So, for this reason, $C=\mathrm{sploc}(y/A)$ is of the form $D_{V,i}(\overline{a})$, for some finite tuple $\overline{a}$ of $A$, and as a set, it is definable over $A$. Also, by the above construction, $q(C)$ is definable over $E^{S}(\Sigma,q(A))$. ∎

The next proposition follows from the proof of Proposition 4.6. We will need this independent result later on, and so we will state it for future reference. We remark that this proposition does not require that the models be $\omega$-saturated. First we introduce the following notation.

Let $\mathbf{q}:=\left<\mathbf{D},\mathbf{S},q\right>$ and $\mathbf{q^{\prime}}:=\left<\mathbf{D^{\prime}},\mathbf{S}^{\prime},q^{\prime}\right>$ be models of $\mathrm{Th}(\mathbf{p})$. Suppose that there is a partial isomorphism between $\mathbf{q}$ and $\mathbf{q}^{\prime}$, represented by the partial isomorphism $\xi:D\dashrightarrow D^{\prime}$ with finitely generated domain $U\subseteq D$, and an embedding $\sigma:L\rightarrow F^{\prime}$, where $L$ is the field generated over $F_{0}$ by the coordinates of the images of the points in $U$.

Observe that a set of the form $D_{V,i}(\overline{a})$ is a subset of $D^{n}$ (for some $n$) which is definable with parameters (given by $\overline{a}$). In what follows, when we write $\xi(\mathrm{sploc}(\overline{x}/U))$, what we mean is $D^{\prime}_{V,i}(\xi(\overline{a}))$, where $D_{V,i}(\overline{a})=\mathrm{sploc}(\overline{x}/U)$.

Proposition 4.8**.**

Let $x\in D_{V,i}\subseteq D$. Then for every tuple $\overline{g}=(g_{1},\ldots,g_{n})$ of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ there exists $x^{\prime}\in D^{\prime}_{V,i}\subseteq D^{\prime}$ such that $(g_{1}x^{\prime},\ldots,g_{n}x^{\prime})\in\xi\left(\mathrm{sploc}(g_{1}x,\ldots,g_{n}x/U)\right)$ and $\left(q^{\prime}(g_{1}x^{\prime}),\ldots,q^{\prime}(g_{n}x^{\prime})\right)$ realises $\sigma\left(\mathrm{qftp}\left((q(g_{1}x),\ldots,q(g_{n}x))/L\right)\right)$.

In the following lemmas, we will preserve the notation used in the definition of special locus.

Lemma 4.9**.**

If $\overline{\alpha}$ is a subtuple of $\overline{g}$ and $y\in X^{+}$ is such that $\overline{g}y\in\mathrm{sploc}(\overline{g}x/A)$, for some $x\in X^{+}$ and $A\subseteq X^{+}$, then $\overline{\alpha}y\in\mathrm{sploc}(\overline{\alpha}x/A)$.

Proof.

Let $\overline{\beta}$ be the complement subtuple of $\overline{\alpha}$ in $\overline{g}$, so that $\overline{g}=(\overline{\alpha},\overline{\beta})$. The result follows from noting that

[TABLE]

∎

Lemma 4.10**.**

Let $\overline{g}_{1},\ldots,\overline{g}_{n}$ be tuples of elements of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$. Let $y_{1},\ldots,y_{n}\in D$, $A\subseteq D$ and let $C:=\mathrm{sploc}(y_{1},\ldots,y_{n}/A)$. Let $\epsilon:C\rightarrow D^{r}$ be given by $(x_{1},\ldots,x_{n})\mapsto(\overline{g}_{1}x_{1},\ldots,\overline{g}_{n}x_{n})$, where $r$ is the sum of the lengths of the $\overline{g}_{i}$. Then the image of $\epsilon$ equals $\mathrm{sploc}(\overline{g}_{1}y_{1}\ldots,\overline{g}_{n}y_{n}/A)$.

Proof.

First we make a quick observation about $X^{+}$. If $X^{+}_{V,i}\subseteq\left(X^{+}\right)^{n}$ is a special domain and $g_{1},\ldots,g_{n}\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$, then the image of the map:

[TABLE]

is again a special domain. This is because we can use a straightforward adaptation of the proof of Lemma 3.22.

Now we return to the setting of the Lemma. We know that the special locus has the form $C=D_{V,i}(\overline{a})$ for some tuple of elements of $A$. Let $m$ be the length of $\overline{a}$. Before tackling the full version of the Lemma, let us focus on the following special case. Let $g_{1},\ldots,g_{n}$ be elements of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$. We will prove that the map $\epsilon_{0}:C\rightarrow D^{n}$ given by $(x_{1},\ldots,x_{n})\mapsto(g_{1}x_{1},\ldots,g_{n}x_{n})$ is a bijection between $C$ and $C^{\prime}:=\mathrm{sploc}((g_{1}x_{1},\ldots,g_{n}x_{n})/A)$. Clearly $\epsilon_{0}$ is injective. Let

[TABLE]

By the above observation on $X^{+}$, we know that the image of $\epsilon_{0}^{\prime}$ is a special domain, which shows that the image of $\epsilon_{0}$ contains $C^{\prime}$. But now we can repeat the argument backwards to show that $C$ is contained in $\epsilon_{0}^{-1}(C^{\prime})$. And so $\epsilon(C)=C^{\prime}$.

Now we prove the general case. Consider the map

[TABLE]

The image of $\epsilon^{\prime}$ is a special domain of $D^{r+m}$ (because of our above observation on $X^{+}$). Let $D_{W,j}$ be the image of $\epsilon^{\prime}$. This shows that the image of $\epsilon$ is of the form $D_{W,j}(\overline{a})$, and therefore $\mathrm{sploc}(\overline{g}_{1}y_{1}\ldots,\overline{g}_{n}y_{n}/A)$ is contained in the image of $\epsilon$.

Say $D_{W^{\prime},j}(\overline{a}^{\prime}):=\mathrm{sploc}(\overline{g}_{1}y_{1}\ldots,\overline{g}_{n}y_{n}/A)$. Observe that the previous paragraph shows that every point of $D_{W^{\prime},j}(\overline{a}^{\prime})$ is of the form $(\overline{g}_{1}x_{1}\ldots,\overline{g}_{n}x_{n})$, for some $(x_{1},\ldots,x_{n})\in C$. Let $\mathrm{pr}$ be the coordinate projection such that $(\overline{g}_{1}y_{1}\ldots,\overline{g}_{n}y_{n},\overline{a})\in D_{W^{\prime},j}$ gets mapped to $(g_{1}y_{1},\ldots,g_{n}y_{n},\overline{a}^{\prime})$. Let $D_{V^{\prime},i}$ be the special domain contained in $\mathrm{pr}(D_{W^{\prime},j})$ such that $(g_{1}y_{1},\ldots,g_{n}y_{n},\overline{a}^{\prime})\in D_{V^{\prime},i}$, with $g_{i}$ being the first coordinate of the tuple $\overline{g}_{i}$. Thanks to the special case, we now know that there is an explicit bijection between $C$ and $\mathrm{sploc}((g_{1}y_{1},\ldots,g_{n}y_{n})/A)$, which finishes the proof. ∎

Corollary 4.11**.**

For any tuple $\overline{g}$ of elements of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ and any $x\in D$, $\mathrm{sploc}(\overline{g}x/x)=\left\{\overline{g}x\right\}$

Lemma 4.12**.**

Given $x_{1},\ldots,x_{n}\in D$ and $A\subseteq D$ we have that

[TABLE]

where $G^{\mathrm{ad}}(\mathbb{Q})^{+}A$ denotes the union of orbits of elements in $A$.

Proof.

Let $D_{V,i}\left(\overline{a}\right)=\mathrm{sploc}\left(\overline{x}/A\right)$. If $m$ is the length of the tuple $\overline{a}$, then consider the tuple $(e,\ldots,e,g_{1},\ldots,g_{m})$. The map $\epsilon:D_{V,i}\rightarrow D_{W,j}$ defined by that element is a bijection (because the statement that says that it is a bijection between the corresponding special domains of $X^{+}$ is a first-order sentence in the theory of $X^{+}$). ∎

We finish this section by describing the types in models of $\mathrm{Th}(\mathbf{p})$. By Proposition 4.6, it is enough to describe the quantifier-free types. First let us set some notation. Let $\mathbf{q}\models\mathrm{Th}(\mathbf{p})$, $x_{1},\ldots,x_{m}\in D$ and $A\subseteq D$. If $v_{1},\ldots,v_{m}$ are first-order variables in $\mathcal{L}_{D}$, we will write

[TABLE]

to mean that $\mathrm{sploc}\left((\overline{g}v_{1},\ldots,\overline{g}v_{m})/A\right)$ is the set of all the formulas in $\mathrm{qftp}\left(\overline{x}/A\right)$ which are either the form $(\overline{g}v_{1},\ldots,\overline{g}v_{m})\in D_{V,i}(\overline{a})$ or of the form $\neg\left((\overline{g}v_{1},\ldots,\overline{g}v_{m})\in D_{V,i}(\overline{a})\right)$.

Proposition 4.13**.**

Let $\mathbf{q}\models\mathrm{Th}(\mathbf{p})$, $x_{1},\ldots,x_{m}\in D$, and let $A\subseteq D$ be closed under the action of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$. Then $\mathrm{qftp}\left(\overline{x}/A\right)$ is determined by:

[TABLE]

where the union is taken over all tuples $\overline{g}=(g_{1},\ldots,g_{n})$ of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$, and $L$ is the field generated over $F_{0}$ by the coordinates of the points in $q(A)$.

Proof.

Let $y_{1},\ldots,y_{m}$ be a realisation of $Q(\overline{v})$ in some elementary extension of $\mathbf{q}$ that also satisfies SF. We need to show that $\mathrm{qftp}(\overline{x}/U_{\mathbf{D}})=\mathrm{qftp}(\overline{y}/U_{\mathbf{D}})$.

The non-trivial atomic formulae in $\mathrm{qftp}(\overline{x}/A)$ that can be written in the domain language $\mathcal{L}_{D}$ are of the form: $x_{i}=a$, $x_{i}=gx_{j}$, $(\overline{g}x_{1},\ldots,\overline{g}x_{m})\in D_{V,i}(\overline{a})$, and their negations, for all parameters $a$ and $\overline{a}$ from $A$, and all $g$, $\overline{g}=(g_{1},\ldots,g_{n})$ from $G^{\mathrm{ad}}(\mathbb{Q})^{+}$. The quantifier-free formulas in the language $\mathcal{L}_{F}$ in $\mathrm{qftp}(\overline{x}/A)$ are already contained in $Q(\overline{v})$.

Observe that formulae of the forms $x_{i}=a$ or $x_{i}=gx_{j}$ are in fact equivalent to a formula of the form $(\overline{g}x_{1},\ldots,\overline{g}x_{m})\in D_{V,i}(\overline{a})$, using the MOD conditions of the special subvarieties $Z_{(e,e)}$ and $Z_{(e,g)}$. So we only need to focus on this last case.

Note that $(\overline{g}x_{1},\ldots,\overline{g}x_{m})\in D_{V,i}(\overline{a})$, for some $\overline{g}=(g_{1},\ldots,g_{n})$ from $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ and some tuple $\overline{a}$ from $A$, if and only if $\mathrm{sploc}\left((\overline{g}x_{1},\ldots,\overline{g}x_{m})/A\right)\subseteq D_{V,i}(\overline{a})$. So $(\overline{g}x_{1},\ldots,\overline{g}x_{m})\in D_{V,i}(\overline{a})$ implies $(\overline{g}y_{1},\ldots,\overline{g}y_{m})\in D_{V,i}(\overline{a})$. The converse works the same way. ∎

5. Models With Standard Fibres

From its construction, we know that the fibres of $p:X^{+}\rightarrow S(\mathbb{C})$ consist of exactly one $\Gamma$-orbit. As $\Gamma$ is infinite, it seems unlikely at first sight that this property about the fibres of $p$ can be expressed with a first-order sentence in $\mathcal{L}$. This is verified once we consider the following partial types:

1. (1)

Let $z\in S(\mathbb{C})$ be a non-Hodge-generic point, that is $\mathrm{spcl}(z)=V\subsetneq S(\mathbb{C})$. For $V$, there are (countably) infintely many corresponding special domains $X^{+}_{V,i}\subset X^{+}$. So the set of formulas (in the variable $v$ with parameter $z$):

[TABLE]

is finitely satisfiable, and so it is part of a complete type. Therefore it is realised in a saturated model of $\mathrm{Th}(\mathbf{p})$. Observe that if $x$ is a realisation of such a type, then we will see discrepancies between special closures on the domain and special closure on the variety, specifically: $\mathrm{spcl}(q(x))\subsetneq q(\mathrm{spcl}(x))$. In this case, we will say that $x$ is a non-standard point. 2. (2)

Even if the domain of a Shimura structure consists only of standard points, we can still have that the fibres of $q$ consists of many $\Gamma$-orbits. To see this, choose $x\in X^{+}$ non-special, let $D_{V,i}=\mathrm{spcl}(x)$, and consider the set of formulas (with variable $v$ and parameter $x$):

[TABLE]

That this set is finitely satisfiable, is a consequence of the fact that, for every $z\in V$, $p^{-1}(z)\cap X^{+}_{V,i}$ is made up of exactly one $\Gamma^{H}$-orbit, where $\Gamma^{H}=\Gamma\cap H(\mathbb{Q})_{+}$, and $H=\mathrm{MT}(x)$.

We will deal with the presence non-standard points and new standard points when we study Shimura covers. In this section we will instead omit these types as we will restrict to the class of models of $\mathrm{Th}(\mathbf{p})$ that satisfy the standard fibres condition, which is the $\mathcal{L}_{\omega_{1},\omega}$-sentence:

[TABLE]

Definition**.**

Given $\mathbf{q}\models\mathrm{Th}(\mathbf{p})$, a point $x\in D$ is called a standard point if $q\left(\mathrm{spcl}(x)\right)=\mathrm{spcl}(q(x))$. A standard section of $q$ is a section $A\subseteq D$ that only contains standard points and such that if $z\in A$, then for every $g\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$ there is $h\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$ such that $hz\in A$ and $q(hz)=q(gz)$.

Remark 5.1*.*

We point out that by $\mathrm{MOD}_{\overline{g},V,i}^{2}$, if $q:D\rightarrow S(F)\models\mathrm{Th}(\mathbf{p})$, then every $z\in S(F)$ has a standard point in its $q$-fibre. Also, is $x\in D$ is standard, then $gx$ is standard for every $g\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$.

Let $\mathrm{Th}_{\mathrm{SF}}(\mathbf{p}):=\mathrm{Th}(\mathbf{p})\cup SF$. As the next Proposition shows, we can construct models of $\mathrm{Th}_{\mathrm{SF}}(\mathbf{p})$ from models of $\mathrm{Th}(\mathbf{p})$.

Lemma 5.2**.**

Given any model $\mathbf{q}\models\mathrm{Th}(\mathbf{p})$, there is $\mathbf{q}^{\prime}\prec\mathbf{q}$ such that $S(F)=S(F^{\prime})$ and $\mathbf{q}^{\prime}\models\mathrm{SF}$.

Proof.

First observe that if $x\in D$ is standard, then $gx$ is standard for any $g\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$. Now choose a standard section of $q$, call it $A$. Standard sections exist because we can look at the space of orbits of $D$ modulo $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ and first choose orbits which contain standard points, and then from each of these orbits, choose representatives modulo $\Gamma$. Let $D^{\prime}=\Gamma A$ and let $q^{\prime}$ be the restriction of $q$ to $D^{\prime}$. Define $\mathbf{q}^{\prime}$ as the structure given by $q^{\prime}:D^{\prime}\rightarrow S(F)$, where $D^{\prime}_{V,i}=D_{V,i}\cap\left(D^{\prime}\right)^{n}$.

The action of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ restricts to an action on $D^{\prime}$ because of the definition of standard section. No $D_{V,i}^{\prime}$ can be empty because the special points of $D$ are in $D^{\prime}$. Indeed, this is because the special domains of a special point in $S(F)$ are all singletons.

It is clear that $\mathbf{q}^{\prime}$ satisfies SS and, because the $D_{V,i}^{\prime}$ are just restrictions of the $D_{V,i}$, we have that $\mathbf{q}^{\prime}$ satisfies $\mathrm{T}(\mathbf{p})$. So $\mathbf{q}^{\prime}\models\mathrm{Th}_{\mathrm{SF}}(\mathbf{p})$ by Proposition 4.6. ∎

When our models have standard fibres, we can strengthen the result from Proposition 4.13 to give a better description of the types.

Proposition 5.3**.**

Let $\mathbf{q}\models\mathrm{Th}_{\mathrm{SF}}(\mathbf{p})$, $x_{1},\ldots,x_{m}\in D$, and let $A\subseteq D$ be closed under the action of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$. Then $\mathrm{qftp}(\overline{x}/A)$ is determined (in $\mathrm{Th}_{\mathrm{SF}}(\mathbf{p})$) by:

[TABLE]

where the union is taken over all tuples $\overline{g}=(g_{1},\ldots,g_{n})$ of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$, and $L$ is the field generated over $F_{0}$ by the coordinates of the points in $q(A)$.

Proof.

Let $y_{1},\ldots,y_{m}$ be a realisation of $Q(\overline{v})$ in some elementary extension of $\mathbf{q}$ that also satisfies SF. By Proposition 4.13 is will suffice to show that $\mathrm{sploc}\left((\overline{g}y_{1},\ldots,\overline{g}y_{m})/A\right)=\mathrm{sploc}\left((\overline{g}x_{1},\ldots,\overline{g}x_{m})/A\right)$.

Note that $(\overline{g}x_{1},\ldots,\overline{g}x_{m})\in D_{V,i}(\overline{a})$, for some $\overline{g}=(g_{1},\ldots,g_{n})$ from $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ and some tuple $\overline{a}$ from $A$, if and only if $\mathrm{sploc}\left((\overline{g}x_{1},\ldots,\overline{g}x_{m})/A\right)\subseteq D_{V,i}(\overline{a})$. So $(\overline{g}x_{1},\ldots,\overline{g}x_{m})\in D_{V,i}(\overline{a})$ implies $(\overline{g}y_{1},\ldots,\overline{g}y_{m})\in D_{V,i}(\overline{a})$. Conversely, suppose $(\overline{g}y_{1},\ldots,\overline{g}y_{m})\in D_{V,i}(\overline{a})$. By hypothesis and SF, there is $\overline{\gamma}\in\Gamma^{\ell m+d}$, where $\ell$ is the length of $\overline{g}$ and $d$ is the length of $\overline{a}$, such that $\overline{\gamma}(\overline{g}x_{1},\ldots,\overline{g}x_{m},\overline{a})=(\overline{g}y_{1},\ldots,\overline{g}y_{m},\overline{a})$. So $(\overline{g}x_{1},\ldots,\overline{g}x_{m},\overline{a})\in D_{V,k}$, for some $k$. But that means that $\mathrm{sploc}\left((\overline{g}x_{1},\ldots,\overline{g}x_{m})/A\right)\subseteq D_{V,k}(\overline{a})$. As special domains of the same special subvariety are either the same or disjoint, we conclude that $D_{V,i}=D_{V,k}$. ∎

Remark 5.4*.*

Let $x_{1},\ldots,x_{m}\in D$ be in different $G^{\mathrm{ad}}(\mathbb{Q})^{+}$-orbits, and let $\overline{g}=(e,g_{1},\ldots,g_{n})$ be a tuple of distinct elements of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$. Let $L$ be the field generated over $F_{0}$ with the coordinates of the $q(x_{i})$, and let $V=\mathrm{spcl}\left(q(x_{1}),\ldots,q(x_{m})\right)$. As we have said before, $\mathrm{qftp}\left(\left(q(\overline{g}x_{1}),\ldots,q(\overline{g}x_{n})\right)/L)\right)$ is equivalent to the minimal algebraic subset of $Z_{\overline{g}}^{V,i}$ defined over $L$ that contains the tuple. This is a subset of the fibre over $\left(q(x_{1}),\ldots,q(x_{m})\right)$ of the morphism

[TABLE]

In fact, it is the $\mathrm{Aut}(\mathbb{C}/L)$-orbit in this fibre containing this tuple.

6. Shimura Covers

From a model theoretic point of view, $\mathrm{Th}(\mathbf{p})$ may not be the most natural first-order theory one can associate with a Shimura variety. For instance, given $S(F)\models T$, it is not immediately clear if one can find an $\mathcal{L}_{D}$-structure $D$ and a map $q:D\rightarrow S(F)$ so that this becomes a model of $\mathrm{Th}(\mathbf{p})$. We address this question by first taking a seemingly different approach to defining Shimura structures.

6.1. The covering sort

Set $\Gamma_{\overline{g}}:=g_{1}^{-1}\Gamma g_{1}\cap\cdots\cap g_{n}^{-1}\Gamma g_{n}$, where $\overline{g}=(g_{1},\ldots,g_{n})$ is a tuple of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$. The quotients of $X^{+}$ by the groups $\Gamma_{\overline{g}}$ form an inverse system of varieties, where the morphisms are those induced by the inclusion of groups. Let $\widehat{S}(\mathbb{C})$ denote the inverse limit of this system. Denote an equivalence class of $\Gamma_{\overline{g}}\backslash X^{+}$ by $[\cdot]_{\overline{g}}$, with the implicit understanding that if $\Gamma_{\overline{g}}=\Gamma_{\overline{h}}$, then $[\cdot]_{\overline{g}}=[\cdot]_{\overline{h}}$. So a point in $\widehat{S}(\mathbb{C})$ can be thought of as a collection of points indexed by tuples $\overline{g}$, such that $\left[x_{\overline{g}}\right]_{\overline{g}}\in\Gamma_{\overline{g}}\backslash X^{+}$ and satisfying that if $\Gamma_{\overline{g}}\subseteq\Gamma_{\overline{h}}$, then $\psi_{\overline{g},\overline{h}}\left(\left[x_{\overline{g}}\right]_{\overline{g}}\right)=\left[x_{\overline{h}}\right]_{\overline{h}}$.

We can define an action of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ on $\widehat{S}(\mathbb{C})$ as the action of $\alpha\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$ on $X^{+}$ induces a map

[TABLE]

which, as a map of algebraic varieties, is defined over $E^{\mathrm{ab}}$. If we let $\overline{g}\alpha^{-1}:=(g_{1}\alpha^{-1},\ldots,g_{n}\alpha^{-1})$, then we can describe the action on components by:

[TABLE]

We cite the following Lemma with the observation that its proof can still be used for the case of mixed Shimura varieties.

Lemma 6.1** (see [10, Lemma 4.3]).**

The group $\Gamma_{\infty}:=\bigcap_{\overline{g}}\Gamma_{\overline{g}}$, where the intersection ranges over all finite tuples of distinct elements of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$, is contained in $Z_{G}(\mathbb{Q})$, where $Z_{G}$ stands for the centre of $G$.

Lemma 6.2**.**

There is an embedding of $X^{+}$ into $\widehat{S}(\mathbb{C})$ given by:

[TABLE]

which is $G^{\mathrm{ad}}(\mathbb{Q})^{+}$-equivariant.

Proof.

Equivariance is immediate from the definition of the action of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ on $\widehat{S}(\mathbb{C})$. Let $x,y\in X^{+}$ be such that $\iota(x)=\iota(y)$. Let $\overline{g}=(e,g_{1},\ldots,g_{n})$. We know from §3.6 that there is an isomorphism:

[TABLE]

Therefore we have $\left(p(x),p(g_{1}x),\ldots,p(g_{n}x)\right)=\left(p(y),p(g_{1}y),\ldots,p(g_{n}y)\right)$, which means that $y\in\Gamma_{\overline{g}}x$. As this happens for every tuple of the form $\overline{g}$, we get that $y\in\Gamma_{\infty}x$, so that $y=x$ by Lemma 6.1. ∎

Let $\widehat{p}:\widehat{S}(\mathbb{C})\rightarrow S(\mathbb{C})$ be given by $\widehat{p}\left(\left(\left[x_{\overline{g}}\right]_{\overline{g}}\right)_{\overline{g}}\right)=p(x_{e})$. Thus we get a commutative diagram:

{X^{+}}$${\widehat{S}(\mathbb{C})}$${S(\mathbb{C})}$$\scriptstyle{p}$$\scriptstyle{\iota}$$\scriptstyle{\widehat{p}}

where $\iota$ is the embedding given in Lemma 6.2.

Now we will use a new form of notation for the points of $\widehat{S}(\mathbb{C})$. From the construction, it is immediate that we can see $\widehat{S}(\mathbb{C})$ as the inverse limit of the system of special subvarieties $Z_{\overline{g}}$, for all tuples $\overline{g}$ from $G^{\mathrm{ad}}(\mathbb{Q})^{+}$. So a point $x\in\widehat{S}(\mathbb{C})$ can be written as $x=\left(z_{\overline{g}}\right)_{\overline{g}}$, where $z_{\overline{g}}\in Z_{\overline{g}}$, and whenever $\overline{g}\subseteq\overline{h}$,

[TABLE]

This structure comes equipped with natural maps $\widehat{p}_{\overline{g}}:\widehat{S}(\mathbb{C})\rightarrow Z_{\overline{g}}$, which are determined by the natural map $\widehat{p}:\widehat{S}(\mathbb{C})\rightarrow S(\mathbb{C})$ given by $\widehat{p}_{\overline{g}}\left(\widetilde{x}\right)=z_{\overline{g}}$.

Lemma 6.3**.**

For every $\widetilde{x}\in\widehat{S}(\mathbb{C})$ and any tuple $\overline{g}$ of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$, we have that:

[TABLE]

Proof.

In order to avoid messy notation, we will prove the statement for an arbitrary tuple from $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ of the form $(\alpha_{1},\alpha_{2})$. The general case can be proven using the same argument. Choose $\widetilde{x}\in\widehat{S}(\mathbb{C})$. We denote its components using both systems of notation, i.e. $\widetilde{x}=(z_{\overline{g}})_{\overline{g}}$ (with $z_{\overline{g}}\in Z_{\overline{g}}$), and $\widetilde{x}=\left(\left[x_{\overline{g}}\right]_{\overline{g}}\right)_{\overline{g}}$, where $z_{\overline{g}}=p\left(\overline{g}x_{\overline{g}}\right)$.

On one hand $\widehat{p}_{(\alpha_{1},\alpha_{2})}\left(\widetilde{x}\right)=z_{(\alpha_{1},\alpha_{2})}$. On the other hand, we know that $\alpha_{1}[x_{\overline{g}}]_{\overline{g}}=\left[\alpha_{1}x_{\overline{g}}\right]_{\overline{g}\alpha_{1}^{-1}}$, so in particular $\alpha_{1}\left[x_{\alpha_{1}}\right]_{\alpha_{1}}=\left[\alpha_{1}x_{\alpha_{1}}\right]_{e}$. Therefore $\widehat{p}\left(\alpha_{1}\widetilde{x},\alpha_{2}\widetilde{x}\right)=\left(\left[\alpha_{1}x_{\alpha_{1}}\right]_{e},\left[\alpha_{2}x_{\alpha_{2}}\right]_{e}\right)$.

By compatibility of components, we have that $\left[x_{(\alpha_{1},\alpha_{1})}\right]_{\alpha_{i}}=\left[x_{\alpha_{i}}\right]_{\alpha_{i}}$ for $i=1,2$. This means that there are $\gamma_{1}\in\alpha_{1}^{-1}\Gamma\alpha_{1}$ and $\gamma_{2}\in\alpha_{2}^{-1}\Gamma\alpha_{2}$ such that $x_{\alpha_{1}}=\gamma_{1}x_{(\alpha_{1},\alpha_{2})}$ and $x_{\alpha_{2}}=\gamma_{2}x_{(\alpha_{1},\alpha_{2})}$. So there are $\gamma_{1}^{\prime},\gamma_{2}^{\prime}\in\Gamma$ such that $\alpha_{1}x_{\alpha_{1}}=\gamma_{1}^{\prime}\alpha_{1}x_{(\alpha_{1},\alpha_{2})}$ and $\alpha_{2}x_{\alpha_{2}}=\gamma_{2}^{\prime}\alpha_{2}x_{(\alpha_{1},\alpha_{2})}$, so that $\widehat{p}\left(\alpha_{1}\widetilde{x},\alpha_{2}\widetilde{x}\right)=p\left(\alpha_{1}x_{(\alpha_{1},\alpha_{2})},\alpha_{2}x_{(\alpha_{1},\alpha_{2})}\right)=z_{(\alpha_{1},\alpha_{2})}$. ∎

If $x_{1},\ldots,x_{m}\in\widehat{S}(\mathbb{C})$, we define the map $\widehat{p}_{\overline{g}}(x_{1},\ldots,x_{m}):=\left(\widehat{p}_{\overline{g}}(x_{1}),\ldots,\widehat{p}_{\overline{g}}(x_{m})\right)$. Given a special subvariety $V$ of $S(\mathbb{C})^{m}$, we will define the set $\widehat{S}(\mathbb{C})_{V,i}$ as the set of points $x\in\widehat{S}^{m}$ such that for every tuple $\overline{g}$ from $G^{\mathrm{ad}}(\mathbb{Q})^{+}$, $\widehat{p}_{\overline{g}}(x)\in Z_{\overline{g}}^{V,i}$.

And so, we get a two-sorted structure $\widehat{\mathbf{p}}:=\left<\widehat{S}(\mathbb{C}),S(\mathbb{C}),\left\{\widehat{p}_{\overline{g}}\right\}_{\overline{g}}\right>$, with each $\widehat{p}_{\overline{g}}$ denoting a function from $\widehat{S}(\mathbb{C})$ to a corresponding power of $S(\mathbb{C})$. The language for $S(\mathbb{C})$ is the same language $\mathcal{L}_{F}$ from before, but now the language on $\widehat{S}(\mathbb{C})$ consists only of relation symbols for each $\widehat{S}(\mathbb{C})_{V,i}$, without an explicit action from $G^{\mathrm{ad}}(\mathbb{Q})^{+}$.

Definition** (Shimura covers).**

A covering structure for a Shimura variety $p:X^{+}\rightarrow S(\mathbb{C})$ is defined in the following way.

1. (a)

Let $\mathcal{L}_{\mathrm{cov}}$ be the language consisting of relation symbols for every $\widehat{S}(\mathbb{C})_{V,i}$, which can be seen as a reduct of the language $\mathcal{L}_{D}$. Let $\widetilde{\mathcal{L}}$ be the language $\widetilde{\mathcal{L}}:=\mathcal{L}_{\mathrm{cov}}\cup\mathscr{L}_{F}\cup\left\{q_{\overline{g}}\right\}$, where every $q_{\overline{g}}$ is a function symbol, indexed by a tuple $\overline{g}$ from $G^{\mathrm{ad}}(\mathbb{Q})^{+}$. 2. (b)

Let $T=\mathrm{Th}(S(\mathbb{C}))$ as before, and let $\widetilde{T}(\widehat{\mathbf{p}})$ be the complete first-order theory of of the two-sorted structure $\widehat{\mathbf{p}}$ in the language $\widetilde{\mathcal{L}}$. Models of $\widetilde{T}$ will be called Shimura covers. 3. (c)

Models of $T$ will usually be denoted simply by $M$, and sometimes we will be more explicit by saying that $M=S(F)$ for some algebraically closed field $F$. Models of $\widetilde{T}(\widehat{\mathbf{p}})$ will be denoted $\widetilde{\mathbf{q}}:=\left<\widetilde{M},M,\left\{q_{\overline{g}}\right\}\right>$ . Given a special subvariety $V$ of $M$, let $\widetilde{M}_{V,i}$ denote a subset of the corresponding power of $\widetilde{M}$. These sets will still be called special domains. 4. (d)

Given an $\widetilde{\mathcal{L}}$-structure $\left<\widetilde{M},M,\left\{q_{\overline{g}}\right\}\right>$ and $B\subseteq\widetilde{M}$, let $\widehat{B}$ denote the set of points in $M$ of the form $q_{g}(b)$, for all $b\in B$ and all $g\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$.

The expectation now is that given $S(F)\models\mathrm{Th}(S(\mathbb{C}))$, we can construct $\widehat{S}(F)$ as the inverse limit of the $Z_{\overline{g}}(F)$, and then the corresponding structure $\left<\widehat{S}(F),S(F),\left\{q_{\overline{g}}\right\}\right>$ will be a model of $\widetilde{T}(\widehat{\mathbf{p}})$. There are, however, a few important details that we need to consider before proceeding.

On one hand, it is true that for every special point $z\in S(\mathbb{C})$, the special domains $\widehat{S}(\mathbb{C})_{z,i}$ consist of exactly one point, which is also true of the special domains $X_{z,i}^{+}$. However, if $p(x)=z$, then Theorem 3.6 says that there is some $\alpha\in G^{\mathrm{ad}}(\mathbb{Q})^{+}$ such that $x$ is the unique fixed point of $\alpha$. This means then that the intersection of special domains $X_{Z_{(e,\alpha)},1}^{+}\cap X^{+}_{Z_{(e,e)},1}$ consists of exactly one point: $(x,x)$. But the intersection $\widehat{S}(\mathbb{C})_{Z_{(e,\alpha)},1}\cap\widehat{S}(\mathbb{C})_{Z_{(e,e)},1}$ has more than one point, as $\alpha$ now fixes more elements of $\widehat{S}(\mathbb{C})$. In other words, if we view $\widehat{S}(\mathbb{C})$ as an $\mathcal{L}_{D}$-structure, then it is not elementarily equivalent to $X^{+}$.

It makes sense then to consider the $\mathcal{L}_{\mathrm{cov}}$-substructure of $\widehat{S}(\mathbb{C})$ which consists only of standard points, call it $\widetilde{S}(\mathbb{C})$. Given a tuple $\overline{g}$ of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$, let $\widetilde{p}_{\overline{g}}:\widetilde{S}(\mathbb{C})\rightarrow S(\mathbb{C})^{n}$ be the restriction of $\widehat{p}_{\overline{g}}$. We therefore get an $\widetilde{\mathcal{L}}$-structure $\widetilde{\mathbf{p}}:=\left<\widetilde{S}(\mathbb{C}),S(\mathbb{C}),\left\{\widetilde{p}_{\overline{g}}\right\}\right>$. In fact, $\widetilde{S}(\mathbb{C})$ also inherits a $G^{\mathrm{ad}}(\mathbb{Q})^{+}$-action from $\widehat{S}(\mathbb{C})$.

Lemma 6.4**.**

$\widetilde{x}\in\widehat{S}(\mathbb{C})$* is non-standard if and only if there is a tuple $\overline{g}$ of elements of $G^{\mathrm{ad}}(\mathbb{Q})^{+}$ and a standard point $\widetilde{y}\in\mathrm{spcl}(\widetilde{x})$ such that $\widehat{p}(\widetilde{x})=\widehat{p}(\widetilde{y})$ and $\widetilde{p}_{\overline{g}}(\widetilde{x})\notin\mathrm{spcl}(\widehat{p}_{\overline{g}}(\widetilde{y}))$.*

Proof.

Suppose first that $\widetilde{x}$ is non-standard. Let $V=\mathrm{spcl}(\widehat{p}(\widetilde{x}))$ and $\widehat{S}(\mathbb{C})_{W,j}=\mathrm{spcl}(\widetilde{x})$. Then $V\subsetneq W$. For some $i\in\mathbb{N}$ we can find $\widetilde{y}\in\widehat{S}(\mathbb{C})_{V,i}\subsetneq\widehat{S}(\mathbb{C})_{W,j}$ such that $\widehat{p}(\widetilde{y})=\widehat{p}(\widetilde{x})$. As $\widetilde{x}\notin\widehat{S}(\mathbb{C})_{V,i}$, then by definition there must be a tuple $\overline{g}$ such that $\widehat{p}_{\overline{g}}(\widetilde{x})\notin Z_{\overline{g}}^{V,i}$. But by Lemma 4.10, $Z_{\overline{g}}^{V,i}=\mathrm{spcl}(\widehat{p}(\widetilde{y}))$.

Conversely, assume that there exist $\overline{g}$ and $\widetilde{y}$ satisfying the conditions for $\widetilde{x}$. Then $\mathrm{spcl}(\widehat{p}(\widetilde{x}))=\mathrm{spcl}(\widehat{p}(\widetilde{y}))=V$, and as $\widetilde{y}$