The Brauer group of $\mathscr{M}_{1,1}$ over algebraically closed fields of characteristic $2$

TL;DR

This paper determines the Brauer group of the moduli stack of elliptic curves over algebraically closed fields of characteristic 2, showing it is isomorphic to Z/2, and extends the computation to finite fields.

Contribution

It provides the first explicit computation of the Brauer group of ,1 moduli stack in characteristic 2, revealing its structure as Z/2 over algebraically closed fields.

Findings

Brauer group over algebraically closed fields is Z/2

Brauer group over finite fields of characteristic 2 computed

Extends understanding of moduli stacks in characteristic 2

Abstract

We prove that the Brauer group of the moduli stack of elliptic curves $\mathscr{M}_{1,1,k}$ over an algebraically closed field $k$ of characteristic $2$ is isomorphic to $\mathbb{Z}/(2)$. We also compute the Brauer group of $\mathscr{M}_{1,1,k}$ where $k$ is a finite field of characteristic $2$.

Equations

$$\mathrm{Br}{\mathcal{M}}_{1,1,k}=\{\begin{array}{ll}\mathbb{Z}/(12)\oplus \mathbb{Z}/(2)& \text{if }{x}^2+x+1\text{ has a root in }k\\ \mathbb{Z}/(24)& \text{otherwise.}\end{array}$$

$${\mathrm{PGL}}_n({\mathcal{O}}_X)\to {\underline{\mathrm{Aut}}}_{{\mathcal{O}}_X\text{-alg}}({\mathrm{Mat}}_{n\times n}({\mathcal{O}}_X))$$

$${\mathcal{A}}_1{\otimes }_{{\mathcal{O}}_X}{\underline{\mathrm{End}}}_{{\mathcal{O}}_X\text{-mod}}({\mathcal{E}}_1)\simeq {\mathcal{A}}_2{\otimes }_{{\mathcal{O}}_X}{\underline{\mathrm{End}}}_{{\mathcal{O}}_X\text{-mod}}({\mathcal{E}}_2)$$

$$(U,\mathcal{E},\sigma )$$

$$(f,f^{♯}):(U_1,{\mathcal{E}}_1,{\sigma }_1)\to (U_2,{\mathcal{E}}_2,{\sigma }_2)$$

$${\iota }_{(U,\mathcal{E},\sigma )}:{\mathbb{G}}_{m,U}\to {\underline{\mathrm{Aut}}}_{(U,\mathcal{E},\sigma )}$$

$${\alpha }_X^{\prime }:\mathrm{Br}X\to {\mathrm{H}}^2(X,{\mathbb{G}}_{m,X})$$

$$(f,f^{♯}):(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$$

$$\mathrm{Br}X$$

$${\alpha }_X:\mathrm{Br}X\to {\mathrm{Br}}^{\prime }X$$

$$\pi :{\mathcal{M}}_{1,1,S}\to {\mathbb{A}}_S^1$$

$$Y^{2}Z+XYZ=X^3-\frac{36}{t-1728}X{Z}^2-\frac{1}{t-1728}{Z}^3$$

$$E_2^{p,q}={\mathrm{H}}^p(\mathbb{Z}/(2),{\mathrm{H}}_{\acute{\mathrm{e}}\mathrm{t}}^q(U,{\mathbb{G}}_m))⟹{\mathrm{H}}_{\acute{\mathrm{e}}\mathrm{t}}^{p+q}(\mathrm{B}(\mathbb{Z}/(2){)}_U,{\mathbb{G}}_m)$$

$${\mathrm{H}}_{\acute{\mathrm{e}}\mathrm{t}}^2(\mathrm{B}(\mathbb{Z}/(2){)}_U,{\mathbb{G}}_m)\simeq {\mathrm{H}}^2(\mathbb{Z}/(2),{\mathrm{H}}_{\acute{\mathrm{e}}\mathrm{t}}^0(U,{\mathbb{G}}_m))\simeq ({\mathbb{G}}_m(U))/(2)$$

$${\mathrm{Br}}^{\prime }{\mathcal{M}}_{1,1,k}\to {\mathrm{Br}}^{\prime }{\mathcal{M}}_{1,1,k}^{\circ }$$

$$1\to {\mu }_{2^n}\to {\mathbb{G}}_m\stackrel{\times 2^n}{\to }{\mathbb{G}}_m\to 1$$

$$0\to (\mathrm{Pic}\mathcal{M})/(2^n)\to {\mathrm{H}}^2(\mathcal{M},{\mu }_{2^n})\to {\mathrm{H}}^2(\mathcal{M},{\mathbb{G}}_m)[2^n]\to 0$$

$$U:=\mathrm{Spec}k[t,(t(t-1728){)}^{-1}]={\mathbb{A}}_k^1\setminus \{0,1728\}$$

$${\mathcal{M}}^{\circ }$$

$${\mathcal{M}}_Z$$

$${\mathcal{M}}^{\circ }$$

$$j_{!}{j}^{\ast }\mathbf{R}{\pi }_{\ast }{\mu }_{2^n}\to \mathbf{R}{\pi }_{\ast }{\mu }_{2^n}\to i_{\ast }{i}^{\ast }\mathbf{R}{\pi }_{\ast }{\mu }_{2^n}\stackrel{+1}{\to }$$

$${\mathrm{H}}^0({\mathbb{A}}_k^1,j_{!}\mathbf{R}{\pi }_{\ast }^{\circ }{\mu }_{2^n})$$

$${\mathrm{H}}^s({\mathbb{A}}_k^1,\mathbf{R}{\pi }_{\ast }{\mu }_{2^n})$$

$${\mathrm{H}}^s({\mathbb{A}}_k^1,i_{\ast }{i}^{\ast }\mathbf{R}{\pi }_{\ast }{\mu }_{2^n})$$

$$Z:=\mathrm{Spec}k(x_1)⨿\cdots ⨿\mathrm{Spec}k(x_r)$$

$${\mathrm{H}}^s({\mathbb{A}}_k^1,j_{!}{\mu }_{\ell })=\{\begin{array}{ll}0& s\ne 1\\ ({\mu }_{\ell }(k){)}^{\oplus (r-1)}& s=1\end{array}.$$

$$j_{!}{\mu }_{\ell }{|}_U\to {\mu }_{\ell }\to i_{\ast }{i}^{\ast }{\mu }_{\ell }\stackrel{+1}{\to }$$

$${\mathrm{H}}^s({\mathbb{A}}_k^1,j_{!}\mathbf{R}{\pi }_{\ast }^{\circ }{\mu }_{\ell })=\{\begin{array}{ll}0& \text{if }s=0,\\ ({\mu }_{\ell }(k){)}^{\oplus (r-1)}& \text{if }s=1,\\ ({\mu }_{\gcd (n,\ell )}(k){)}^{\oplus (r-1)}& \text{if }s=2.\end{array}$$

$$\mathcal{C}:=j_{!}\mathbf{R}{\pi }_{\ast }^{\circ }{\mu }_{\ell }$$

$${\mathrm{H}}^s({\mathbb{A}}_k^1,{\tau }_{\le t}\mathcal{C})\simeq {\mathrm{H}}^s({\mathbb{A}}_k^1,\mathcal{C})$$

Full text

1. Introduction

Let $\mathscr{M}_{1,1,\mathbb{Z}}$ denote the moduli stack of elliptic curves over $\mathbb{Z}$. For any scheme $S$, we denote by $\mathscr{M}_{1,1,S}:=S\times_{\mathbb{Z}}\mathscr{M}_{1,1,\mathbb{Z}}$ the restriction of $\mathscr{M}_{1,1,\mathbb{Z}}$ to the category of schemes over $S$.

Antieau and Meier [AM16, 11.2] computed the Brauer group $\operatorname{Br}\mathscr{M}_{1,1,S}$ for various base schemes $S$, and in particular proved that for any algebraically closed field $k$ of characteristic not $2$ the Brauer group $\operatorname{Br}\mathscr{M}_{1,1,k}$ is trivial. The purpose of this note is to compute $\operatorname{Br}\mathscr{M}_{1,1,k}$ in the characteristic $2$ case. This then completes the calculation of $\operatorname{Br}\mathscr{M}_{1,1,k}$ over algebraically closed fields $k$. We summarize the result in the following theorem.

Theorem 1.1** ([AM16, 11.2] in $\operatorname{char}k\neq 2$).**

Let $k$ be an algebraically closed field. Then $\operatorname{Br}\mathscr{M}_{1,1,k}$ is [math] unless $\operatorname{char}k=2$, in which case $\operatorname{Br}\mathscr{M}_{1,1,k}=\mathbb{Z}/(2)$.

To prove the theorem, we calculate the cohomology groups $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\mathscr{M}_{1,1,k},\mu_{n})$ for varying $n$. There are essentially two ways to approach this calculation: (1) using the coarse moduli space; (2) using a presentation of $\mathscr{M}_{1,1,k}$ as a quotient stack. In this paper we give a new proof of the Antieau-Meier result using approach (1), and calculate in characteristic 2 using approach (2).

We also compute the Brauer group of $\mathscr{M}_{1,1,k}$ where $k$ is a finite field of characteristic $2$:

Theorem 1.2**.**

Let $k$ be a finite field of characteristic $2$. Then

[TABLE]

An outline of the paper is as follows.

In Section 2 we state definitions and recall general facts about the Brauer group of algebraic stacks.

In Section 3 we record some general remarks regarding $\operatorname{Br}\mathscr{M}_{1,1,S}$. We show that if $S$ is a quasi-compact scheme admitting an ample line bundle and if at least one prime is invertible on $S$, then $\operatorname{Br}\mathscr{M}_{1,1,S}\simeq\operatorname{Br}^{\prime}\mathscr{M}_{1,1,S}$. The restriction of $\mathscr{M}_{1,1,\mathbb{Z}}$ to the dense open substack of elliptic curves $E/S$ with $j$-invariant $j(E)\in\Gamma(S,\mathcal{O}_{S})$ for which $j(E)$ and $j(E)-1728$ are invertible is a trivial $\mathbb{Z}/(2)$-gerbe over the coarse space $\mathbb{A}_{\mathbb{Z}}^{1}\setminus\{0,1728\}$, and we use this fact to conclude that $\operatorname{Br}\mathscr{M}_{1,1,k}$ is a subgroup of $\mathbb{Z}/(2)\oplus\mathbb{Z}/(2)$ for an algebraically closed field $k$ of arbitrary characteristic.

In Section 4 we give a second proof of Antieau and Meier’s result above (that $\operatorname{Br}\mathscr{M}_{1,1,k}=0$ if $k=\overline{k}$ and $\operatorname{char}k\neq 2$). Using a dévissage argument, we study the relationship between the cohomology of $\mu_{n}$ on the stack $\mathscr{M}_{1,1,k}$ and on $\mathbb{A}_{k}^{1}$, in terms of the stabilizer groups of elliptic curves with $j$-invariant $0,1728\in\mathbb{A}_{k}^{1}$. This may be of independent interest for computing the Brauer groups of other separated Deligne-Mumford stacks whose coarse moduli space is a smooth curve over an algebraically closed field with vanishing Picard group.

In Section 5 we prove Theorem 1.1 and Theorem 1.2. Antieau and Meier suggest in [AM16, 11.3] that the characteristic $2$ case can be settled using the $\operatorname{GL}_{2}(\mathbb{Z}/(3))$-cover $Y(3)\to\mathscr{M}_{1,1,k}$, where $Y(3)$ denotes the moduli stack of elliptic curves with full level 3 structure, and indeed we use this presentation $\mathscr{M}_{1,1,k}\simeq[Y(3)/\operatorname{GL}_{2}(\mathbb{Z}/(3))]$ as a global quotient stack to show that its Brauer group is in fact nonzero. We use the “Hesse presentation” of $Y(3)$ as in [FO10]; it is shown in Appendix A that this presentation coincides with the usual Weierstrass presentation as in [KM85]. The cohomological descent spectral sequence associated to the covering $Y(3)\to\mathscr{M}_{1,1,k}$ reduces our task to a computation of the first group cohomology of a $6$-dimensional representation of $\operatorname{GL}_{2}(\mathbb{Z}/(3))$ over $\mathbb{F}_{2}$.

1.3 (Acknowledgements).

I thank my advisor Martin Olsson for suggesting this research topic and for his generosity in sharing his ideas. I am also grateful to Benjamin Antieau, Siddharth Mathur, and Lennart Meier for helpful discussions. During this project, I received support from the Raymond H. Sciobereti Fellowship.

2. The Brauer group of algebraic stacks

Let $(X,\mathcal{O}_{X})$ be a locally ringed site [Gir71, V, §4], [Sta18, 04EU]. For any quasi-coherent $\mathcal{O}_{X}$-module $\mathscr{E}$, we set $\operatorname{GL}(\mathscr{E}):=\underline{\operatorname{Aut}}_{\mathcal{O}_{X}\text{-mod}}(\mathscr{E})$ and let $\operatorname{PGL}(\mathscr{E})$ be the sheaf quotient of $\operatorname{GL}(\mathscr{E})$ by $\mathbb{G}_{m,X}$ via the diagonal embedding. We denote $\operatorname{GL}_{n}(\mathcal{O}_{X}):=\operatorname{GL}(\mathcal{O}_{X}^{\oplus n})$ and $\operatorname{PGL}_{n}(\mathcal{O}_{X}):=\operatorname{PGL}(\mathcal{O}_{X}^{\oplus n})$. A basic fact about these groups is the Skolem-Noether theorem, which states that the morphism

[TABLE]

is an isomorphism (see [Gir71, V.4.1]).

Definition 2.1 (Azumaya algebras).

[Gro68a, §2], [Gir71, V, §4] Let $(X,\mathcal{O}_{X})$ be a locally ringed site. An Azumaya $\mathcal{O}_{X}$-algebra is a quasi-coherent (non-commutative, unital) $\mathcal{O}_{X}$-algebra $\mathscr{A}$ such that there exists a covering $\{X_{i}\to X\}_{i\in I}$, positive integers $n_{i}$, and $\mathcal{O}_{X_{i}}$-algebra isomorphisms $\mathscr{A}|_{X_{i}}\simeq\operatorname{Mat}_{n_{i}\times n_{i}}(\mathcal{O}_{X_{i}})$.

Two Azumaya algebras $\mathscr{A}_{1}$ and $\mathscr{A}_{2}$ are Morita equivalent if there exist finite type locally free $\mathcal{O}_{X}$-modules $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$, everywhere of positive rank, and an isomorphism

[TABLE]

of $\mathcal{O}_{X}$-algebras. Under tensor product of Azumaya algebras, Morita equivalence classes of Azumaya algebras form an abelian group $\operatorname{Br}X$ called the (Azumaya) Brauer group of $X$ in which $[\mathscr{A}]^{-1}=[\mathscr{A}^{\operatorname{op}}]$ and the identity element is the class of trivial Azumaya algebras $[\underline{\operatorname{End}}_{\mathcal{O}_{X}\textnormal{-mod}}(\mathcal{E})]$.

Definition 2.2 (Gerbe of trivializations).

[Gir71, IV, §4.2], [Ols16, 12.3.5] There is a natural way to associate, to every Azumaya $\mathcal{O}_{X}$-algebra $\mathscr{A}$, a $\mathbb{G}_{m,X}$-gerbe $\mathcal{G}_{\mathscr{A}}$ called the gerbe of trivializations of $\mathscr{A}$. An object of $\mathcal{G}_{\mathscr{A}}$ is a triple

[TABLE]

consisting of an object $U\in X$, a finite type locally free $\mathcal{O}_{U}$-module $\mathcal{E}$ (necessarily everywhere positive rank), and an isomorphism $\sigma:\underline{\operatorname{End}}_{\mathcal{O}_{U}\textnormal{-mod}}(\mathcal{E})\to\mathscr{A}|_{U}$ of $\mathcal{O}_{U}$-algebras. A morphism

[TABLE]

consists of a morphism $f\in\operatorname{Mor}_{X}(U_{1},U_{2})$ and an isomorphism $f^{\sharp}:f^{\ast}\mathcal{E}_{2}\to\mathcal{E}_{1}$ of $\mathcal{O}_{U_{1}}$-modules such that $\sigma_{2}=\sigma_{1}\circ\rho_{f^{\sharp}}$ where $\rho_{f^{\sharp}}$ denotes conjugation by $f^{\sharp}$. For any object $(U,\mathcal{E},\sigma)\in\mathcal{G}_{\mathscr{A}}$, there is a canonical injection

[TABLE]

of sheaves on $X/U$, sending $u\mapsto(\operatorname{id}_{U},u)$; this is in fact an isomorphism, since if $(\operatorname{id}_{U},f^{\sharp})\in\operatorname{Aut}_{\mathcal{G}_{\mathscr{A}}(U)}((U,\mathcal{E},\sigma))$ then $f^{\sharp}\in Z(\operatorname{End}_{\mathcal{O}_{U}\textnormal{-mod}}(\mathcal{E}))$, which coincides with $\mathcal{O}_{U}$ since $Z(\operatorname{Mat}_{n\times n}(A))=A$ for any commutative, unital ring $A$.

By the Skolem-Noether theorem, any two local trivializations of $\mathscr{A}$ are locally related by an automorphism of the trivializing vector bundle $\mathcal{E}$, i.e. any two objects of $\mathcal{G}_{\mathscr{A}}$ are locally isomorphic. Furthermore, according to the definition, an Azumaya algebra is locally trivial, i.e. for any $U\in X$ there exists a covering $\{U_{i}\to U\}$ such that the fiber category $\mathcal{G}_{\mathscr{A}}(U_{i})$ is nonempty. These considerations show that $\mathcal{G}_{\mathscr{A}}$ is a $\mathbb{G}_{m,X}$-gerbe.

The assignment $\mathscr{A}\mapsto\mathcal{G}_{\mathscr{A}}$ induces a group homomorphism

[TABLE]

which is injective since a $\mathbb{G}_{m,X}$-gerbe $\mathcal{G}$ is trivial if and only if $\mathcal{G}(X)$ is nonempty.

For a morphism

[TABLE]

of locally ringed sites, the diagram

[TABLE]

is commutative.

Lemma 2.3**.**

Let $\mathscr{X}$ be a $\mathbb{G}_{m,X}$-gerbe over a locally ringed site $X$. The class $[\mathscr{X}]\in\mathrm{H}^{2}(X,\mathbb{G}_{m,X})$ is in the image of $\alpha_{X}^{\prime}$ if and only if $\mathscr{X}$ admits a $1$-twisted finite locally free sheaf of everywhere positive rank.

The usual proof (c.f. [dJ03, 2.14], [Lie08, 3.1.2.1], [Ols16, 12.3.11]) of Lemma 2.3 applies more generally to the case of $\mathbb{G}_{m}$-gerbes over an arbitrary locally ringed site.

We will only consider locally ringed sites $(X,\mathcal{O}_{X})$ whose underlying site $X$ is quasi-compact [Sta18, 090G]. For such $X$, the Brauer group $\operatorname{Br}X$ is a torsion group.

Definition 2.4.

The torsion subgroup of $\mathrm{H}^{2}(X,\mathbb{G}_{m,X})$, denoted $\operatorname{Br}^{\prime}X$, is called the cohomological Brauer group and the restriction

[TABLE]

of $\alpha_{X}^{\prime}$ to $\operatorname{Br}^{\prime}X$ is called the Brauer map.

We will consider algebraic stacks using the étale topology except in Section 5 (the case of characteristic $2$) in which we will require the flat topology.

Surjectivity of the Brauer map may be checked on a finite flat surjective covering (c.f. [Gab78, II, Lemma 4], [dJ03, 2.15], [Lie08, 3.1.3.5]):

Proposition 2.5**.**

Let $f:X\to Y$ be a finitely presented, finite, flat, surjective morphism of algebraic stacks. A class $\beta\in\mathrm{H}^{2}(Y,\mathbb{G}_{m,Y})$ is in the image of $\alpha_{Y}^{\prime}$ if and only if its pullback $f^{\ast}\beta\in\mathrm{H}^{2}(X,\mathbb{G}_{m,X})$ is in the image of $\alpha_{X}^{\prime}$.

Proof.

Let $\mathscr{Y}$ be the $\mathbb{G}_{m,Y}$-gerbe corresponding to $\beta$. Set $\mathscr{X}:=X\times_{Y}\mathscr{Y}$ and let $F:\mathscr{X}\to\mathscr{Y}$ be the induced morphism of algebraic stacks. If $\mathscr{X}$ is in the image of $\alpha_{X}^{\prime}$, then there exists a $1$-twisted finite locally free $\mathcal{O}_{\mathscr{X}}$-module $\mathscr{E}$ of everywhere positive rank. The pushforward $F_{\ast}\mathscr{E}$ is a $1$-twisted, finite locally free $\mathcal{O}_{\mathscr{Y}}$-module of everywhere positive rank. Hence $\mathscr{Y}$ is in the image of $\alpha_{Y}^{\prime}$.

The other direction follows from commutativity of the diagram 2.2.2. ∎

Corollary 2.6**.**

Let $f:X\to Y$ be a finitely presented, finite, flat, surjective morphism of algebraic stacks. If $\alpha_{X}$ is an isomorphism, then $\alpha_{Y}$ is an isomorphism.

Corollary 2.7**.**

Let $X$ be a smooth separated generically tame Deligne-Mumford stack over a field $k$ with quasi-projective coarse moduli space. Then the Brauer map $\alpha_{X}$ is surjective.

Proof.

By Kresch-Vistoli [KV04, 2.1,2.2], such $X$ has a finite flat surjection $Z\to X$ where $Z$ is a quasi-projective $k$-scheme. By Gabber’s theorem (see [dJ03, 1.1]), the Brauer map is surjective for $Z$. Thus the Brauer map is surjective for $X$ by Proposition 2.5. ∎

Remark 2.8.

If $\operatorname{char}k\neq 2$, the stack $\mathscr{M}_{1,1,k}$ is generically tame and so Corollary 2.7 implies surjectivity of the Brauer map $\alpha_{\mathscr{M}_{1,1,k}}$. For the case $\operatorname{char}k=2$, see Lemma 3.1.

3. Preliminary observations

The purpose of this section is to prove Lemma 3.4 below. Let us start, however, with a few preliminary observations about the stack $\mathscr{M}_{1,1}$ and its Brauer group.

The stack $\mathscr{M}_{1,1,\mathbb{Z}}$ is a Deligne-Mumford stack smooth and separated over $\mathbb{Z}$ [Ols16, 13.1.2]; hence if $S$ is a regular Noetherian scheme then $\mathscr{M}_{1,1,S}$ is a regular Noetherian stack. For any locally Noetherian scheme $S$, the morphism

[TABLE]

sending an elliptic curve to its $j$-invariant identifies $\mathbb{A}_{S}^{1}$ with the coarse moduli space of $\mathscr{M}_{1,1,S}$ [FO10, 4.4].

In general, if $\mathscr{X}$ is a separated Deligne-Mumford stack and $\pi:\mathscr{X}\to X$ is its coarse moduli space, then $\pi$ is initial among maps from $\mathscr{X}$ to an algebraic space, so the map $X(\mathcal{G})\to\mathscr{X}(\mathcal{G})$ is an isomorphism for any group scheme $\mathcal{G}$; moreover if $U\to X$ is an etale morphism, then $\pi_{U}:\mathscr{X}\times_{X}U\to U$ is a coarse moduli space. Applying these observations to $\mathcal{G}=\mathbb{G}_{a},\mathbb{G}_{m},\mu_{n}$ implies that the canonical maps $\mathcal{O}_{X}\to\pi_{\ast}\mathcal{O}_{\mathscr{X}}$, $\mathbb{G}_{m,X}\to\pi_{\ast}\mathbb{G}_{m,\mathscr{X}}$, $\mu_{n,X}\to\pi_{\ast}\mu_{n,\mathscr{X}}$ are isomorphisms; thus we will omit subscripts and denote $\mu_{n},\mathbb{G}_{m}$ for the corresponding sheaves on either $\mathscr{M}_{1,1,S}$ or $\mathbb{A}_{S}^{1}$.

Lemma 3.1**.**

Let $S$ be a quasi-compact scheme admitting an ample line bundle, and suppose that at least one prime $p$ is invertible in $S$. Then the Brauer map $\alpha_{\mathscr{M}_{1,1,S}}:\operatorname{Br}\mathscr{M}_{1,1,S}\to\operatorname{Br}^{\prime}\mathscr{M}_{1,1,S}$ is an isomorphism.

Proof.

By [KM85, 4.7.2], for $N\geq 3$ the moduli stack of full level $N$ structures is representable by an affine $\mathbb{Z}[\frac{1}{N}]$-scheme $Y(N)$. Set $Y(N)_{S}:=Y(N)\times_{\mathbb{Z}[\frac{1}{N}]}S$; the projection $Y(N)_{S}\to S$ is an affine morphism, hence $Y(N)_{S}$ is quasi-compact and admits an ample line bundle, hence the Brauer map $\alpha_{Y(N)_{S}}$ is surjective by Gabber’s theorem (see [dJ03]), and, since the map $Y(N)_{S}\to\mathscr{M}_{1,1,S}$ is finite locally free, we have by Corollary 2.6 that $\alpha_{\mathscr{M}_{1,1,S}}$ is surjective. ∎

Lemma 3.2**.**

Let $U:=\operatorname{Spec}\mathbb{Z}[t,(t(t-1728))^{-1}]\subset\mathbb{A}_{\mathbb{Z}}^{1}$ and let $\mathscr{M}_{1,1,\mathbb{Z}}^{\circ}:=U\times_{\mathbb{A}_{\mathbb{Z}}^{1}}\mathscr{M}_{1,1,\mathbb{Z}}$. Then the restriction $\pi^{\circ}:\mathscr{M}_{1,1,\mathbb{Z}}^{\circ}\to U$ of $\pi$ to $U$ is a trivial $\mathbb{Z}/(2)$-gerbe, i.e. $\mathscr{M}_{1,1,\mathbb{Z}}^{\circ}\simeq\mathrm{B}(\mathbb{Z}/(2))_{U}$.

Proof.

Let $S$ be a scheme and let $E_{1},E_{2}$ be two elliptic curves over $S$. If $j(E_{1})=j(E_{2})\in\Gamma(S,\mathcal{O}_{S})$ and $j(E_{i}),j(E_{i})-1728$ are units of $\Gamma(S,\mathcal{O}_{S})$, then by [Del75, 5.3] one can find a finite étale cover $S^{\prime}\to S$ such that there is an isomorphism $S^{\prime}\times_{S}E_{1}\simeq S^{\prime}\times_{S}E_{2}$ of elliptic curves over $S^{\prime}$. For any connected scheme $S$ and an elliptic curve $E/S$ for which $j(E)$ and $j(E)-1728$ are invertible, we have $\operatorname{Aut}(E/S)\simeq\mathbb{Z}/(2)$ by [KM85, (8.4.2)]. It suffices now to show that there is an elliptic curve $E_{U}$ over $U$ with $j$-invariant $t$. For this we may take the elliptic curve $E_{U}$ defined by the Weierstrass equation

[TABLE]

which satisfies $\Delta(E_{U})=\frac{t^{2}}{(t-1728)^{3}}$ and $j(E_{U})=t$ (see [Sil09, Proposition III.1.4(c)]). ∎

Lemma 3.3**.**

Let $k$ be an algebraically closed field and let $U$ be a smooth curve over $k$. If $\operatorname{Pic}(U)=0$, then $\operatorname{Br}^{\prime}\mathrm{B}(\mathbb{Z}/(2))_{U}\simeq(\mathbb{G}_{m}(U))/(2)$.

Proof.

The cohomological descent spectral sequence associated to the cover $U\to\mathrm{B}(\mathbb{Z}/(2))_{U}$ is of the form

[TABLE]

with differentials $\operatorname{E}_{2}^{p,q}\to\operatorname{E}_{2}^{p+2,q-1}$. We have by [Mil80, III.2.22 (d)] that $\mathrm{H}_{\mathrm{\acute{e}t}}^{q}(U,\mathbb{G}_{m})=0$ for all $q\geq 2$. Moreover, we have $\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(U,\mathbb{G}_{m})=\operatorname{Pic}(U)=0$ by assumption. Thus the only row of the $\operatorname{E}_{2}$-page of 3.3.1 containing nonzero entries is $q=0$, which gives an isomorphism

[TABLE]

of abelian groups. ∎

Lemma 3.4**.**

Let $k$ be an algebraically closed field. If $\operatorname{char}k\neq 2,3$, then $\operatorname{Br}^{\prime}\mathscr{M}_{1,1,k}$ is a subgroup of $\mathbb{Z}/(2)\oplus\mathbb{Z}/(2)$. If $\operatorname{char}k$ is $2$ or $3$, then $\operatorname{Br}^{\prime}\mathscr{M}_{1,1,k}$ is a subgroup of $\mathbb{Z}/(2)$.

Proof.

We have that $\mathscr{M}_{1,1,k}$ is regular Noetherian and that $\mathscr{M}_{1,1,k}^{\circ}:=\mathscr{M}_{1,1,\mathbb{Z}}^{\circ}\times_{\mathbb{Z}}k$ is a dense open substack; thus by [AM16, 2.5(iv)] the map

[TABLE]

induced by restriction is an injection. Here Lemma 3.2 implies $\operatorname{Br}^{\prime}\mathscr{M}_{1,1,k}^{\circ}=\operatorname{Br}^{\prime}\mathrm{B}(\mathbb{Z}/(2))_{U}$ for $U=\operatorname{Spec}k[t,(t(t-1728))^{-1}]$, and Lemma 3.3 implies $\operatorname{Br}^{\prime}\mathrm{B}(\mathbb{Z}/(2))_{U}$ is $\mathbb{Z}/(2)\oplus\mathbb{Z}/(2)$ if $\operatorname{char}k\neq 2,3$ and $\mathbb{Z}/(2)$ otherwise (here we use that $k^{\times}=(k^{\times})^{2}$ since $k$ is algebraically closed). ∎

4. The case $\operatorname{char}k$ is not $2$

Antieau and Meier [AM16] compute the Brauer group $\operatorname{Br}\mathscr{M}_{1,1,S}$ for various base schemes $S$, including algebraically closed fields $k$ of odd characteristic [AM16, 11.2] (the case $\operatorname{char}k\neq 2$ in Theorem 1.1). In this section we give a proof via a dévissage argument, using the fact that the coarse moduli space morphism $\pi:\mathscr{M}\to\mathbb{A}_{k}^{1}$ is a trivial $\mathbb{Z}/(2)$-gerbe away from $0,1728\in\mathbb{A}_{k}^{1}$ (see Lemma 3.2). Our proof is divided into two cases, depending on whether $\operatorname{char}k=3$ or $\operatorname{char}k\neq 3$ (this will determine whether we puncture $\mathbb{A}_{k}^{1}$ at one or two points, respectively). We first fix notation and record some observations that apply to both cases.

4.1.

We abbreviate $\mathscr{M}:=\mathscr{M}_{1,1,k}$. By Lemma 3.1, the Brauer map $\alpha_{\mathscr{M}}:\operatorname{Br}\mathscr{M}\to\operatorname{Br}^{\prime}\mathscr{M}$ is an isomorphism. By Lemma 3.4, the main task is to show that the $2$-torsion in $\operatorname{Br}\mathscr{M}$ is [math].

For any integer $n\geq 1$, the étale Kummer sequence

[TABLE]

gives an exact sequence

[TABLE]

of abelian groups. Since we have $\operatorname{Pic}\mathscr{M}\simeq\mathbb{Z}/(12)$ by [FO10], we wish to compute $\mathrm{H}^{2}(\mathscr{M},\mu_{2^{n}})$.

Set

[TABLE]

with inclusion $j:U\to\mathbb{A}_{k}^{1}$ and let $i:Z\to\mathbb{A}_{k}^{1}$ be the complement with reduced induced closed subscheme structure. (Thus, if $\operatorname{char}k$ is $2$ or $3$ then $Z\simeq\operatorname{Spec}k$, otherwise $Z\simeq\operatorname{Spec}k\amalg\operatorname{Spec}k$.) Set

[TABLE]

with projections $\pi^{\circ}:\mathscr{M}^{\circ}\to U$ and $\pi_{Z}:\mathscr{M}_{Z}\to Z$. We have a commutative diagram

[TABLE]

with cartesian squares.

We have a distinguished triangle

[TABLE]

in the derived category of bounded-below complexes of abelian sheaves on the étale site of $\mathbb{A}_{k}^{1}$, whose associated long exact sequence has the form

[TABLE]

since $j^{\ast}\mathbf{R}\pi_{\ast}\mu_{2^{n}}\simeq\mathbf{R}\pi^{\circ}_{\ast}\mu_{2^{n}}$ and

[TABLE]

for all $s$. We will first compute the groups $\mathrm{H}^{s}(\mathbb{A}_{k}^{1},j_{!}j^{\ast}\mathbf{R}\pi_{\ast}\mu_{2^{n}})$ in the left column of 4.1.4.

Lemma 4.2**.**

Let $k$ be an algebraically closed field, let $x_{1},\dotsc,x_{r}\in\mathbb{A}_{k}^{1}$ be $r$ distinct $k$-points, set

[TABLE]

and let $U=\mathbb{A}_{k}^{1}\setminus Z$ be the complement with inclusion $j:U\to\mathbb{A}_{k}^{1}$. For any positive integer $\ell$ invertible in $k$, we have

[TABLE]

Proof.

Let $i:Z\to\mathbb{A}_{k}^{1}$ be the inclusion. We have a distinguished triangle

[TABLE]

in the derived category of bounded-below complexes of abelian sheaves on the big étale site of $\mathbb{A}_{k}^{1}$, which gives a long exact sequence

{\mathrm{H}^{0}(\mathbb{A}_{k}^{1},j_{!}\mu_{\ell}|_{U})}$${\mathrm{H}^{0}(\mathbb{A}_{k}^{1},\mu_{\ell})}$${\mathrm{H}^{0}(Z,\mu_{\ell})}$${\mathrm{H}^{1}(\mathbb{A}_{k}^{1},j_{!}\mu_{\ell}|_{U})}$${\mathrm{H}^{1}(\mathbb{A}_{k}^{1},\mu_{\ell})}$${\mathrm{H}^{1}(Z,\mu_{\ell})}$${\mathrm{H}^{2}(\mathbb{A}_{k}^{1},j_{!}\mu_{\ell}|_{U})}$${\mathrm{H}^{2}(\mathbb{A}_{k}^{1},\mu_{\ell})}$${\mathrm{H}^{2}(Z,\mu_{\ell})}$${\mathrm{H}^{3}(\mathbb{A}_{k}^{1},j_{!}\mu_{\ell}|_{U})}$${\dotsb}

in cohomology. The map $\mathrm{H}^{0}(\mathbb{A}_{k}^{1},\mu_{\ell})\to\mathrm{H}^{0}(Z,\mu_{\ell})$ is identified with the diagonal map $\mu_{\ell}(k)\to(\mu_{\ell}(k))^{\oplus r}$. Since $k$ is algebraically closed, the etale site of $Z$ is trivial, hence $\mathrm{H}^{s}(Z,\mu_{\ell})=0$ for $s\geq 1$. By [Del77, Exp. 1, III, (3.6)] we have $\mathrm{H}^{s}(\mathbb{A}_{k}^{1},\mu_{\ell})=0$ for $s\geq 2$. We have $\mathbb{G}_{m}(\mathbb{A}_{k}^{1})\simeq\mathbb{G}_{m}(k)$ and the multiplication-by-$\ell$ map $\times\ell:\mathbb{G}_{m}(k)\to\mathbb{G}_{m}(k)$ is surjective; thus $\mathrm{H}^{1}(\mathbb{A}_{k}^{1},\mu_{\ell})=\mathrm{H}^{1}(\mathbb{A}_{k}^{1},\mathbb{G}_{m})[\ell]=(\operatorname{Pic}\mathbb{A}_{k}^{1})[\ell]=0$ by the Kummer sequence. ∎

Lemma 4.3**.**

In the setup of Lemma 4.2, let $n$ be any positive integer and let $\pi^{\circ}:\mathrm{B}(\mathbb{Z}/(n))_{U}\to U$ be the trivial $\mathbb{Z}/(n)$-gerbe over $U$. Then

[TABLE]

Proof.

We set

[TABLE]

for convenience. We will compute the groups $\mathrm{H}^{s}(\mathbb{A}_{k}^{1},\mathcal{C})$ using the fact that the canonical truncations $\tau_{\leq s}\mathcal{C}$ satisfy

[TABLE]

for $s\leq t$. For any $s\in\mathbb{Z}$, the distinguished triangle

[TABLE]

gives a long exact sequence

[TABLE]

where

[TABLE]

since $j_{!}$ is exact.

Since $\pi^{\circ}:\mathrm{B}(\mathbb{Z}/(n))_{U}\to U$ is a trivial $\mathbb{Z}/(n)$-gerbe, by Lemma B.1 we have

[TABLE]

where $\mu_{\ell}[n]$ and $\mu_{\ell}/(n)$ are defined by the exact sequence

[TABLE]

of abelian sheaves. Since $k$ is algebraically closed of characteristic prime to $\ell$, the sheaves $\mu_{\ell}[n]$ and $\mu_{\ell}/(n)$ are both isomorphic to $\mu_{\gcd(n,\ell)}$, but for us the difference is important for reasons of functoriality (as $\ell$ is allowed to vary). More precisely, if $\ell_{1}$ divides $\ell_{2}$, then the inclusion $\mu_{\ell_{1}}\to\mu_{\ell_{2}}$ induces an inclusion

[TABLE]

whereas

[TABLE]

is not necessarily injective since an element $x\in\mu_{\ell_{1}}$ which is not an $n$th power of any $y_{1}\in\mu_{\ell_{1}}$ may be an $n$th power of some $y_{2}\in\mu_{\ell_{2}}$ (in particular, if $\ell_{2}=n\ell_{1}$, then 4.3.5 is the zero morphism).

We have

[TABLE]

since $\pi^{\circ}$ is a coarse moduli space morphism and $\mathbf{R}^{1}\pi^{\circ}_{\ast}\mu_{\ell}\simeq\mu_{\gcd(n,\ell)}$ by 4.3.4. Applying Lemma 4.2 to the case $s=1$ in 4.3.3 implies $\mathrm{H}^{0}(\mathbb{A}_{k}^{1},\tau_{\leq 1}\mathcal{C})=0$ and gives isomorphisms $\mathrm{H}^{1}(\mathbb{A}_{k}^{1},j_{!}\mu_{\ell})\simeq\mathrm{H}^{1}(\mathbb{A}_{k}^{1},\tau_{\leq 1}\mathcal{C})$ and $\mathrm{H}^{2}(\mathbb{A}_{k}^{1},\tau_{\leq 1}\mathcal{C})\simeq\mathrm{H}^{1}(\mathbb{A}_{k}^{1},j_{!}\mu_{\gcd(n,\ell)})$.

Since $\mathbf{R}^{2}\pi^{\circ}_{\ast}\mu_{\ell}\simeq\mu_{\gcd(n,\ell)}$ by 4.3.4 and $\mathrm{H}^{s}(\mathbb{A}_{k}^{1},j_{!}\mu_{\gcd(n,\ell)})=0$ for $s=-2,-1,0$, the case $s=2$ in 4.3.3 gives isomorphisms $\mathrm{H}^{s}(\mathbb{A}_{k}^{1},\tau_{\leq 1}\mathcal{C})\simeq\mathrm{H}^{s}(\mathbb{A}_{k}^{1},\tau_{\leq 2}\mathcal{C})$ for $s=0,1,2$, which implies the desired result. ∎

4.4 (Proof of Theorem 1.1 for $\operatorname{char}k=3$).

If $\operatorname{char}k=3$, then $Z$ consists of one point, so taking $r=1$ in Lemma 4.3 implies

[TABLE]

for $s=0,1,2$. Therefore, to compute $\mathrm{H}^{2}(\mathscr{M},\mu_{2^{n}})$, it now remains to compute $\mathrm{H}^{2}(Z,i^{\ast}\mathbf{R}\pi_{\ast}\mu_{2^{n}})$ in 4.1.4. The stabilizer of any object of $\mathscr{M}$ of lying over $i:Z\to\mathbb{A}_{k}^{1}$ is the automorphism group of an elliptic curve with $j$-invariant [math], which is the semidirect product $\Gamma=\mathbb{Z}/(3)\rtimes\mathbb{Z}/(4)$ since $k$ has characteristic $3$. The underlying reduced stack $(\mathscr{M}_{Z})_{\operatorname{red}}$ is the residual gerbe associated to the unique point of $|\mathscr{M}_{Z}|$ and is isomorphic to the classifying stack $\mathrm{B}\Gamma_{k}$. We have natural isomorphisms

[TABLE]

where isomorphism 1 follows from proper base change [Ols05, 1.3], isomorphism 2 is by invariance of étale site for nilpotent thickenings and the fact that $2^{n}$ is invertible on $\mathscr{M}_{Z}$, and isomorphism 3 is by the cohomological descent spectral sequence for the covering $\operatorname{Spec}k\to\mathrm{B}\Gamma_{k}$ (and the fact that $\mathrm{H}^{i}(\operatorname{Spec}k,\mu_{2^{n}})=0$ for $i>0$ since $k$ is algebraically closed). The Hochschild-Serre spectral sequence for the exact sequence

[TABLE]

gives an isomorphism

[TABLE]

where $\mathrm{H}^{i}(\mathbb{Z}/(3),\mu_{2^{n}}(k))=0$ for $i>0$ since $3$ is coprime to the order of $\mu_{2^{n}}(k)$. Since the first term in the last row of the diagram 4.1.4 is zero by 4.4.1, the above observations imply that we have natural inclusions

[TABLE]

compatible with the inclusions $\mu_{2^{n}}\subset\mu_{2^{n+1}}$ for all $n$. The inclusion $\mu_{2^{n}}\subset\mu_{2^{n+2}}$ induces the zero map $\mu_{2^{n}}(k)/(4)\to\mu_{2^{n+2}}(k)/(4)$, so $\mathrm{H}^{2}(\mathscr{M},\mu_{2^{n}})\to\mathrm{H}^{2}(\mathscr{M},\mu_{2^{n+2}})$ is the zero map as well, hence

[TABLE]

which by 4.1.1 gives $\mathrm{H}^{2}(\mathscr{M},\mathbb{G}_{m})[2^{n}]=0$ for all $n$.

4.5 (Proof of Theorem 1.1, for $\operatorname{char}k\neq 2,3$).

We describe the terms in 4.1.4. For the right column, we have

[TABLE]

by [ACV03, A.0.7]. For the middle column, we have

[TABLE]

since $\mathbb{A}_{k}^{1}$ is the coarse moduli space of $\mathscr{M}$, and we have

[TABLE]

where isomorphism 1 follows since $k^{\times}=(k^{\times})^{2^{n}}$, isomorphism 2 is by [Mum65], and isomorphism 3 holds for $n\gg 0$. For the left column, we have

[TABLE]

by Lemma 4.3.

To summarize, 4.1.4 simplifies to

[TABLE]

for $n\gg 0$, and counting the number of elements in each group in 4.5.1 implies that the last morphism

[TABLE]

is injective. Furthermore, the inclusion

[TABLE]

induces the zero map

[TABLE]

so the map $\mathrm{H}^{2}(\mathscr{M},\mu_{2^{n}})\to\mathrm{H}^{2}(\mathscr{M},\mu_{2^{n+2}})$ is the zero map as well, hence

[TABLE]

which by 4.1.1 gives $\mathrm{H}^{2}(\mathscr{M},\mathbb{G}_{m})[2^{n}]=0$ for all $n$.

5. The case $\operatorname{char}k$ is $2$

In this section we prove Theorem 1.1 (in case $\operatorname{char}k=2$) and Theorem 1.2. For convenience, we denote $\operatorname{GL}_{n,p}:=\operatorname{GL}_{n}(\mathbb{Z}/(p))$ and $\operatorname{SL}_{n,p}:=\operatorname{SL}_{n}(\mathbb{Z}/(p))$. We denote by $\mathsf{e}$ the identity element of $\operatorname{GL}_{n,p}$.

5.1 (Hesse presentation of $\mathscr{M}_{1,1,k}$).

By [FO10, 6.2] (and explained in more detail in A.6), there is a left action of $\operatorname{GL}_{2,3}$ on the $\mathbb{Z}[\frac{1}{3}]$-algebra

[TABLE]

sending

[TABLE]

for which the corresponding right action of $\operatorname{GL}_{2,3}$ on the $\mathbb{Z}[\frac{1}{3}]$-scheme

[TABLE]

gives a presentation

[TABLE]

of $\mathscr{M}_{1,1,\mathbb{Z}[\frac{1}{3}]}$ as a global quotient stack. The morphism

[TABLE]

is given by the elliptic curve

[TABLE]

over $S_{\mathrm{H}}$.

5.2 (Cohomological descent).

Let $k$ be an algebraically closed field of characteristic $2$. The Brauer map $\alpha_{\mathscr{M}_{1,1,k}}:\operatorname{Br}\mathscr{M}_{1,1,k}\to\operatorname{Br}^{\prime}\mathscr{M}_{1,1,k}$ is an isomorphism by Lemma 3.1. By Lemma 3.4, there is only $2$-torsion in $\operatorname{Br}\mathscr{M}_{1,1,k}$. By Grothendieck’s fppf-étale comparison theorem for smooth commutative group schemes [Gro68b, (11.7)], it suffices to compute the $2$-torsion in $\mathrm{H}_{\operatorname{fppf}}^{2}(\mathscr{M}_{1,1,k},\mathbb{G}_{m})$. Since $\operatorname{Spec}k$ is a reduced scheme, we have

[TABLE]

by [FO10, 1.1]. Thus, for any integer $n$, the fppf Kummer sequence

[TABLE]

gives an exact sequence

[TABLE]

of abelian groups. It remains to compute the middle term $\mathrm{H}_{\operatorname{fppf}}^{2}(\mathscr{M}_{1,1,k},\mu_{2})$.

The cohomological descent spectral sequence associated to the cover 5.1.3 is of the form

[TABLE]

with differentials $\operatorname{E}_{2}^{p,q}\to\operatorname{E}_{2}^{p+2,q-1}$.

Let

[TABLE]

be a fixed primitive $3$rd root of unity. By the Chinese Remainder Theorem, there is a $k$-algebra isomorphism

[TABLE]

sending $\mu\mapsto(\nu_{1},\nu_{2})$ and $\omega\mapsto(\xi,\xi^{2})$. Since $S_{\mathrm{H},k}$ is a smooth curve over an algebraically closed field, we have by [Mil80, III.2.22 (d)] that $\mathrm{H}_{\mathrm{\acute{e}t}}^{q}(S_{\mathrm{H},k},\mathbb{G}_{m})=0$ for all $q\geq 2$; since $S_{\mathrm{H},k}$ is a disjoint union of two copies of a distinguished affine open subset of $\mathbb{A}_{k}^{1}$, we have $\mathrm{H}_{\mathrm{\acute{e}t}}^{1}(S_{\mathrm{H},k},\mathbb{G}_{m})=\operatorname{Pic}(S_{\mathrm{H},k})=0$. By [Gro68b, (11.7)] we have $\mathrm{H}_{\operatorname{fppf}}^{q}(S_{\mathrm{H},k},\mathbb{G}_{m})=\mathrm{H}_{\mathrm{\acute{e}t}}^{q}(S_{\mathrm{H},k},\mathbb{G}_{m})$ for all $q\geq 0$; thus the fppf Kummer sequence implies $\mathrm{H}_{\operatorname{fppf}}^{q}(S_{\mathrm{H},k},\mu_{2})=0$ for all $q\geq 2$. Furthermore, we have $\mathrm{H}_{\operatorname{fppf}}^{0}(S_{\mathrm{H},k},\mu_{2})=0$ since $S_{\mathrm{H},k}$ is the product of two integral domains of characteristic $2$. Thus the only nonzero terms on the $\operatorname{E}_{2}$-page of 5.2.3 occur on the $q=1$ row, so we have an isomorphism

[TABLE]

for all $p\geq 0$. We are interested in the case $p=1$.

5.3 (Description of the $\operatorname{GL}_{2,3}$-action on $\mathrm{H}_{\operatorname{fppf}}^{1}(S_{\mathrm{H},k},\mu_{2})$).

We describe the abelian group

[TABLE]

and the left $\operatorname{GL}_{2,3}$-module structure it inherits from 5.1.1. Since $k[\mu,(\mu^{3}-1)^{-1}]$ is a principal localization of the polynomial ring $k[\mu]$ by a polynomial $\mu^{3}-1=(\mu-1)(\mu-\xi)(\mu-\xi^{2})$ splitting into three distinct irreducible factors, we have an isomorphism

[TABLE]

of abelian groups. Thus 5.2.4 and the Kummer sequence 5.2.1 gives an isomorphism

[TABLE]

of abelian groups, with generators given by the classes of $\nu_{i}-\xi^{j}$ for $i=1,2$ and $j=0,1,2$.

The isomorphism 5.2.4 is given by the map

[TABLE]

for $s_{0},s_{1}\in k[\mu,\frac{1}{\mu^{3}-1}]$. The inverse of 5.2.4 is given by the map

[TABLE]

where $f_{i}(\nu_{i})\in k[\nu_{i},\frac{1}{\nu_{i}^{3}-1}]$. (Note that, if we set $\mathrm{A}_{1}(t):=\frac{t}{\xi-\xi^{2}}+\frac{\xi}{\xi-1}$ and $\mathrm{A}_{2}(t):=\frac{-t}{\xi-\xi^{2}}+\frac{-1}{\xi-1}$, then $\mathrm{A}_{i}(\xi^{j})$ is the Kronecker delta function.)

A computation with 5.1.1, 5.3.3, 5.3.4 shows that the action of $\operatorname{GL}_{2,3}$ on the right hand side of 5.2.4 is given by

[TABLE]

for $f_{i}(\nu_{i})\in k[\nu_{i},\frac{1}{\nu_{i}^{3}-1}]$. A computation with 5.3.5 (and using that $\operatorname{char}k=2$) shows that the action of $\operatorname{GL}_{2,3}$ on 5.3.2 is given by 5.3.6, where every element is considered up to multiplication by $k^{\times}$.

[TABLE]

5.4.

We compute $\mathrm{H}^{1}(\operatorname{GL}_{2,3},M)$. (In Appendix C we provide Magma code that can be used to verify this computation.) We have a filtration of groups

[TABLE]

where each is a normal subgroup of the next. Here $\mathrm{Q}_{8}$ denotes the quaternion group

[TABLE]

and is identified with the subgroup of $\operatorname{GL}_{2,3}$ as follows:

[TABLE]

The quotient $\operatorname{GL}_{2,3}/\operatorname{SL}_{2,3}$ is cyclic of order $2$ and is generated by $\mathsf{M}_{1}$ in 5.3.6. The quotient $\operatorname{SL}_{2,3}/\mathrm{Q}_{8}$ is cyclic of order $3$ and is generated by $\mathsf{M}_{2}$ in 5.3.6. For $i=1,2$, let $\langle\mathsf{M}_{i}\rangle$ denote the subgroup of $\operatorname{GL}_{2,3}$ generated by $\mathsf{M}_{i}$. We note that $\operatorname{SL}_{2,3}$ is generated by $\mathsf{i}$ and $\mathsf{M}_{2}$.

Let

[TABLE]

be the forgetful functor. An inspection of 5.3.6 implies that $F(M)$ is the direct sum $N_{1}\oplus N_{2}$ where $N_{i}$ is the $\operatorname{SL}_{2,3}$-submodule of $F(M)$ generated by the classes of $\nu_{i}-1,\nu_{i}-\xi,\nu_{i}-\xi^{2}$, and moreover $\mathsf{M}_{1}$ switches the summands $N_{1}$ and $N_{2}$. Under the adjunction

[TABLE]

the projection map $F(M)\simeq N_{1}\oplus N_{2}\to N_{1}$ onto the first factor corresponds to a morphism

[TABLE]

of $\operatorname{GL}_{2,3}$-modules. Given $m\in M$, write $m=n_{1}+n_{2}$ for $n_{i}\in N_{i}$; then the image of $m$ under 5.4.2 is the function $\varphi_{m}\in\operatorname{Hom}_{\mathbb{Z}[\operatorname{SL}_{2,3}]}(\mathbb{Z}[\operatorname{GL}_{2,3}],N_{1})$ such that $\varphi_{m}([\mathsf{e}])=n_{1}$ and $\varphi_{m}([\mathsf{M}_{1}])=\mathsf{M}_{1}\cdot n_{2}$; thus 5.4.2 is an isomorphism.

A computation using 5.3.6 and the identities

[TABLE]

shows that the action of an element $\mathsf{g}\in\operatorname{SL}_{2,3}$ on $N_{1}$ is by left multiplication by the matrix $T_{\mathsf{g}}$ as in 5.4.4, with elements of $N_{1}$ being viewed as vertical vectors. We note $T_{-\mathsf{e}}=T_{\mathsf{i}}^{2}=T_{\mathsf{j}}^{2}=T_{\mathsf{k}}^{2}=\operatorname{id}_{N_{1}}$, i.e. $-\mathsf{e}$ acts trivially on $N_{1}$.

[TABLE]

Since $M$ is an induced module, the restriction map

[TABLE]

is an isomorphism so we reduce to computing $\mathrm{H}^{1}(\operatorname{SL}_{2,3},N_{1})$.

The Hochschild-Serre spectral sequence for the inclusion $\mathrm{Q}_{8}\trianglelefteq\operatorname{SL}_{2,3}$ degenerates on the $\mathrm{E}_{2}$ page since the order of the quotient group $\langle\mathsf{M}_{2}\rangle$ is coprime to the order of $N_{1}$. In particular the restriction map

[TABLE]

is an isomorphism.

Let $\mathrm{C}^{i}(\mathrm{Q}_{8},N_{1}):=\mathrm{Fun}((\mathrm{Q}_{8})^{i},N_{1})$ denote the group of inhomogeneous $i$-cochains. By Remark 5.5, the group $\operatorname{SL}_{2,3}$ has a natural left action on $\mathrm{C}^{i}(\mathrm{Q}_{8},N_{1})$ (by entrywise conjugation on the source $(\mathrm{Q}_{8})^{i}$ and by its usual action on $N_{1}$) such that the differentials in the inhomogeneous cochain complex

[TABLE]

are $\operatorname{SL}_{2,3}$-linear. Since the order of the subgroup $\langle\mathsf{M}_{2}\rangle$ is coprime to the orders of $\mathrm{C}^{i}(\mathrm{Q}_{8},N_{1})$, we have that $\mathrm{H}^{0}(\langle\mathsf{M}_{2}\rangle,\mathrm{H}^{1}(\mathrm{Q}_{8},N_{1}))\simeq(\mathrm{H}^{1}(\mathrm{Q}_{8},N_{1}))^{\mathsf{M}_{2}}$ is isomorphic to the middle cohomology of the sequence

[TABLE]

i.e. cohomology commutes with taking $\mathsf{M}_{2}$-invariants.

We now describe $\ker((d_{1})^{\mathsf{M}_{2}})$ and $\operatorname{im}((d_{0})^{\mathsf{M}_{2}})$.

An element $f\in(\mathrm{C}^{1}(\mathrm{Q}_{8},N_{1}))^{\mathsf{M}_{2}}$ is a function $f:\mathrm{Q}_{8}\to N_{1}$ satisfying

[TABLE]

for all $\mathsf{g}\in\mathrm{Q}_{8}$. We have that $f\in\ker d_{1}$ if

[TABLE]

for all $\mathsf{g}_{1},\mathsf{g}_{2}\in\mathrm{Q}_{8}$.

Suppose $f\in\ker((d_{1})^{\mathsf{M}_{2}})=(\ker d_{1})\cap(\mathrm{C}^{1}(\mathrm{Q}_{8},N_{1}))^{\mathsf{M}_{2}}$; taking $(\mathsf{g}_{1},\mathsf{g}_{2})=(\mathsf{e},\mathsf{e})$ in 5.4.8 implies $f(\mathsf{e})=0$; taking $\mathsf{g}=-\mathsf{e}$ in 5.4.7 implies that

[TABLE]

for some $s\in\mathbb{Z}/(2)$; taking $(\mathsf{g}_{1},\mathsf{g}_{2})=(-\mathsf{e},-\mathsf{e})$ in 5.4.8 and using the fact that $-\mathsf{e}$ acts trivially on $N_{1}$ implies that $2f(-\mathsf{e})=0$, which imposes no condition on $s$. We note that

[TABLE]

for any $\mathsf{g}\in\operatorname{SL}_{2,3}$.

Setting $\mathsf{g}=\mathsf{i},\mathsf{j},\mathsf{k}$ in 5.4.7 and using 5.4.3 gives

[TABLE]

respectively; thus we have

[TABLE]

for some $s_{1},s_{2},s_{3}\in\mathbb{Z}/(2)$.

Setting either $\mathsf{g}_{1}=-\mathsf{e}$ or $\mathsf{g}_{2}=-\mathsf{e}$ in 5.4.8 implies

[TABLE]

for any $\mathsf{g}\in\mathrm{Q}_{8}$.

Setting $(\mathsf{g}_{1},\mathsf{g}_{2})=(\pm\mathsf{i},\pm\mathsf{j}),(\pm\mathsf{j},\pm\mathsf{k}),(\pm\mathsf{k},\pm\mathsf{i})$ in 5.4.8 (where the signs can vary independently of each other) all impose the condition

[TABLE]

for $s,s_{2}$ (check the case $(\mathsf{g}_{1},\mathsf{g}_{2})=(\mathsf{i},\mathsf{j})$, then use 5.4.10 to show that changing the signs don’t give new relations, then use 5.4.9 to show that one can permute using left multiplication by $\mathsf{M}_{2}$).

Setting $(\mathsf{g}_{1},\mathsf{g}_{2})=(\pm\mathsf{j},\pm\mathsf{i}),(\pm\mathsf{k},\pm\mathsf{j}),(\pm\mathsf{i},\pm\mathsf{k})$ in 5.4.8 (where the signs can vary independently of each other) all impose the condition

[TABLE]

for $s_{3}$ (check the case $(\mathsf{g}_{1},\mathsf{g}_{2})=(\mathsf{j},\mathsf{i})$, then use 5.4.10 to show that changing the signs don’t give new relations, then use 5.4.9 to show that one can permute using left multiplication by $\mathsf{M}_{2}$).

Setting $(\mathsf{g}_{1},\mathsf{g}_{2})=(\pm\mathsf{g},\pm\mathsf{g})$ for $\mathsf{g}=\mathsf{i},\mathsf{j},\mathsf{k}$ (where the signs can vary independently of each other) all impose the condition

[TABLE]

on $s,s_{2},s_{3}$ (check the case $\mathsf{g}=\mathsf{i}$, then use 5.4.10 to show that changing the signs don’t give new relations, then use 5.4.9 to show that one can permute using left multiplication by $\mathsf{M}_{2}$), but 5.4.13 is implied by 5.4.11 and 5.4.12.

These are the only relations satisfied by the $s,s_{1},s_{2},s_{3}$. Thus we have

[TABLE]

since there are no relations on $s,s_{1}\in\mathbb{Z}/(2)$.

An element of $(\mathrm{C}^{0}(\mathrm{Q}_{8},N_{1}))^{\mathsf{M}_{2}}$ corresponds to an element $(t,t,t)\in N_{1}$; since every element of $\operatorname{SL}_{2,3}$ fixes elements of this form (see 5.4.4), the image of $(t,t,t)$ under $(d_{0})^{\mathsf{M}_{2}}$ corresponds to the function $f:\mathrm{Q}_{8}\to N_{1}$ sending every element to $(0,0,0)$, in other words

[TABLE]

which implies

[TABLE]

and so

[TABLE]

by combining 5.4.14 with 5.4.6, 5.4.5, 5.2.5, and 5.2.2. ∎

Remark 5.5 (The inhomogeneous cochain complex admits a left $G$-action).

Let $G$ be a group, let $H\trianglelefteq G$ be a normal subgroup, and let $M$ be a left $G$-module. Set $P_{i}:=\mathbb{Z}[H^{i+1}]$; we denote by $[\mathsf{h}_{0},\dotsc,\mathsf{h}_{i}]$ the canonical $\mathbb{Z}$-basis of $P_{i}$. We view $P_{i}$ as a left $H$-module via the diagonal action $\mathsf{h}\cdot[\mathsf{h}_{0},\dotsc,\mathsf{h}_{i}]=[\mathsf{h}\mathsf{h}_{0},\dotsc,\mathsf{h}\mathsf{h}_{i}]$; then $P_{i}$ is a free left $\mathbb{Z}[H]$-module with basis consisting of elements of the form $[\mathsf{e},\mathsf{h}_{1},\dotsc,\mathsf{h}_{i}]$. Applying the functor $\operatorname{Hom}_{H}(-,M)$ to the bar resolution

[TABLE]

gives the usual homogeneous cochain complex

[TABLE]

whose cohomology gives $\mathrm{H}^{i}(H,M)$.

We note that there is a natural left $G$-action on $\operatorname{Hom}_{\mathbb{Z}[H]}(P_{i},M)$ for which the differential $\delta_{i}:\operatorname{Hom}_{\mathbb{Z}[H]}(P_{i},M)\to\operatorname{Hom}_{\mathbb{Z}[H]}(P_{i+1},M)$ is $G$-linear. Namely, the action of $\mathsf{g}\in G$ on $\varphi_{i}\in\operatorname{Hom}_{\mathbb{Z}[H]}(P_{i},M)$ is described by

[TABLE]

for all $\mathsf{h}_{0},\dotsc,\mathsf{h}_{i}\in H$. Let

[TABLE]

denote the abelian group of functions $H^{i}\to M$. Via the usual abelian group isomorphism

[TABLE]

sending $\varphi_{i}\mapsto\{(\mathsf{h}_{1},\dotsc,\mathsf{h}_{i})\mapsto\varphi_{i}(\mathsf{e},\mathsf{h}_{1},\mathsf{h}_{1}\mathsf{h}_{2},\dotsc,\mathsf{h}_{1}\dotsb\mathsf{h}_{i})\}$, the abelian group $\mathrm{C}^{i}(H,M)$ inherits a left action of $G$ described by

[TABLE]

for $\mathsf{g}\in G$ and $f_{i}\in\mathrm{C}^{i}(H,M)$. The inhomogeneous cochain complex

[TABLE]

is $G$-linear as well.

For $f_{0}\in\mathrm{C}^{0}(H,M)$, we have $(d_{0}f_{0})(\mathsf{h}_{1})=\mathsf{h}_{1}\cdot f_{0}(\mathsf{e})-f_{0}(\mathsf{e})$.

For $f_{1}\in\mathrm{C}^{1}(H,M)$, we have $(d_{1}f_{1})(\mathsf{h}_{1},\mathsf{h}_{2})=\mathsf{h}_{1}\cdot f_{1}(\mathsf{h}_{2})-f_{1}(\mathsf{h}_{1}\mathsf{h}_{2})+f_{1}(\mathsf{h}_{1})$.

Let $\Sigma:=G/H$ be the quotient; then there is an induced left action of $\Sigma$ on the cohomology $\mathrm{h}^{i}(\mathrm{C}^{\bullet}(H,M))$. In case $G\to\Sigma$ has a section, in which case $G$ is the semi-direct product $G\simeq H\rtimes\Sigma$, then this $\Sigma$-action coincides with the one obtained by restricting the $G$-action on $\mathrm{C}^{\bullet}(H,M)$ to $\Sigma$.

Remark 5.6.

The arguments used in 5.3 and 5.4 are similar to those of Mathew and Stojanoska [MS16, Appendix B], who show $\mathrm{H}^{1}(\operatorname{GL}_{2,3},(TMF(3)_{0})^{\times})=\mathbb{Z}/(12)$ where $\operatorname{GL}_{2,3}$ acts on

[TABLE]

as in [Sto14, §4.3].

Note 5.7 (Explicit description of inhomogeneous 1-cocycles).

We describe the 1-cocycles $\operatorname{GL}_{2,3}\to M$ obtained via the compositions 5.4.6 and 5.4.5. By our computation in 5.4, the 1-cocycles

[TABLE]

are of the form

[TABLE]

for some $s,s_{1}\in\mathbb{Z}/(2)$. Suppose

[TABLE]

is a 1-cocycle such that $f_{\operatorname{SL}_{2,3}}$ is fixed by the action of $\mathsf{M}_{2}$ (see 5.5.1) and which satisfies $f_{\operatorname{SL}_{2,3}}(\mathsf{g})=f_{\mathrm{Q}_{8}}(\mathsf{g})$ for $\mathsf{g}\in\mathrm{Q}_{8}$. We have

[TABLE]

for all $\mathsf{g}\in\operatorname{SL}_{2,3}$; taking $\mathsf{g}=\mathsf{M}_{2}$ gives $\mathsf{M}_{2}\cdot f_{\operatorname{SL}_{2,3}}(\mathsf{M}_{2})=f_{\operatorname{SL}_{2,3}}(\mathsf{M}_{2})$. Taking $\mathsf{g}_{1}=\mathsf{g}_{2}=\mathsf{M}_{2}$ in the 1-cocycle condition 5.4.8 then gives $f_{\operatorname{SL}_{2,3}}(\mathsf{M}_{2})=0$. Thus we have

[TABLE]

for any $\mathsf{g}\in\operatorname{SL}_{2,3}$, again by 5.4.8.

By Shapiro’s lemma 5.4.5, there is a 1-cocycle

[TABLE]

such that precomposing with the inclusion $\operatorname{SL}_{2,3}\subset\operatorname{GL}_{2,3}$ and postcomposing with the projection $\mathrm{Ind}_{\operatorname{SL}_{2,3}}^{\operatorname{GL}_{2,3}}(N_{1})\to N_{1}$ gives $f_{\operatorname{SL}_{2,3}}$. After altering $f_{\operatorname{GL}_{2,3}}$ by a 1-coboundary, we may assume by Note 5.8 that $f_{\operatorname{GL}_{2,3}}$ is given by the formula 5.8.1, namely

[TABLE]

for any $i,j\in\{0,1\}$ and $\mathsf{g}\in\operatorname{SL}_{2,3}$. Any element $\mathsf{g}\in\operatorname{GL}_{2,3}$ may be expressed in the form

[TABLE]

where $i_{1}\in\{0,1\}$ and $i_{2}\in\{0,1,2\}$ and $\mathsf{h}\in\mathrm{Q}_{8}$. We have formulas

[TABLE]

and so

[TABLE]

where equality 1 is by 5.7.2 and equality 2 is by 5.7.1 and 5.7.3 and equality 3 is since $\mathsf{M}_{1}^{j}\cdot\mathsf{h}\cdot\mathsf{M}_{1}^{-j}\in\mathrm{Q}_{8}$ (see 5.7.3). This is summarized in 5.7.4 below.

[TABLE]

Note 5.8 (The Shapiro isomorphism and inhomogeneous 1-cocycles).

111Ehud Meir’s MathOverflow post [Mei16] was helpful in working out the details of this section.

Let $G$ be a group, let $H\subseteq G$ be a normal subgroup of finite index such that the projection $G\to G/H$ has a section $G/H\to G$ whose image corresponds to a subgroup $\Sigma$ of $G$. Let $N$ be a left $H$-module and let $\mathrm{Ind}_{H}^{G}N:=\operatorname{Hom}_{\mathbb{Z}[H]}(\mathbb{Z}[G],N)$ denote the associated induced left $G$-module. We recall that the left $G$-action on $\mathrm{Ind}_{H}^{G}N$ sends $\varphi\mapsto\mathsf{g}\varphi$ where $(\mathsf{g}\varphi)(x)=\varphi(x\mathsf{g})$.

We describe the inverse of the Shapiro isomorphism $\mathrm{H}^{1}(G,\mathrm{Ind}_{H}^{G}N)\to\mathrm{H}^{1}(H,N)$ in terms of inhomogeneous cochains. Suppose given a function

[TABLE]

which satisfies

[TABLE]

for all $\mathsf{h}_{1},\mathsf{h}_{2}\in H$. We construct a 1-cocycle

[TABLE]

which restricts to $f$, i.e. satisfies $s(\mathsf{h})(1\cdot[\mathsf{e}])=f(\mathsf{h})$ for all $\mathsf{h}\in H$. Note that every element of $\mathsf{g}\in G$ may be written uniquely in the form

[TABLE]

for $\mathsf{h}\in H$ and $\sigma\in\Sigma$, hence the collection $\{[\sigma]\}_{\sigma\in\Sigma}$ forms a basis for $\mathbb{Z}[G]$ as a left $\mathbb{Z}[H]$-module. We set

[TABLE]

for $\mathsf{h}\in H$ and $\sigma,\xi\in\Sigma$ and extend $\mathbb{Z}[H]$-linearly. Given $\mathsf{g}_{1},\mathsf{g}_{2}\in G$ where $\mathsf{g}_{i}=\mathsf{h}_{i}\sigma_{i}$ with $\mathsf{h}_{i}\in H$ and $\sigma_{i}\in\Sigma$, for any $\xi\in\Sigma$ we have

[TABLE]

and

[TABLE]

and

[TABLE]

which implies

[TABLE]

by $\mathbb{Z}[H]$-linearity and since $f$ is a 1-cocycle; hence $s$ is a 1-cocycle. ∎

5.9 (Proof of Theorem 1.2).

Let $k^{\mathrm{sep}}$ be a fixed separable closure of $k$ and let $\mathrm{G}_{k}:=\mathrm{Gal}(k^{\mathrm{sep}}/k)\simeq\widehat{\mathbb{Z}}$ be the absolute Galois group. Set $\mathscr{M}:=\mathscr{M}_{1,1,k}$ and $\mathscr{M}^{\mathrm{sep}}:=\mathscr{M}_{1,1,k^{\mathrm{sep}}}$. We have $\operatorname{Br}\mathscr{M}=\operatorname{Br}^{\prime}\mathscr{M}$ by Lemma 3.1. The Leray spectral sequence for the map $\mathscr{M}\to\operatorname{Spec}k$ is of the form

[TABLE]

with differentials $\mathrm{E}_{2}^{p,q}\to\mathrm{E}_{2}^{p+2,q-1}$. Here we have $\Gamma(\mathscr{M}^{\mathrm{sep}},\mathbb{G}_{m})=\Gamma(\mathbb{A}_{k^{\mathrm{sep}}}^{1},\mathbb{G}_{m})=(k^{\mathrm{sep}})^{\times}$ since $\mathscr{M}^{\mathrm{sep}}\to\mathbb{A}_{k^{\mathrm{sep}}}^{1}$ is the coarse moduli space map. Since $k$ is a finite field, we have that $\mathrm{H}^{0}_{\mathrm{\acute{e}t}}(\mathscr{M}^{\mathrm{sep}},\mathbb{G}_{m})$ is a torsion group. Moreover $\mathrm{H}^{1}_{\mathrm{\acute{e}t}}(\mathscr{M}^{\mathrm{sep}},\mathbb{G}_{m})\simeq\operatorname{Pic}(\mathscr{M}^{\mathrm{sep}})\simeq\mathbb{Z}/(12)$ is a torsion group by [FO10]. Thus by e.g. [Fu11, 4.3.7] or [GS06, 6.1.3] we have $\mathrm{E}_{2}^{p,q}=0$ for $(p,q)\in\mathbb{Z}_{\geq 2}\times\{0,1\}$. This means there is an exact sequence

[TABLE]

of abelian groups.

By [FO10], we have that $\operatorname{Pic}(\mathscr{M}^{\mathrm{sep}})\simeq\mathbb{Z}/(12)$ is generated by the class of the Hodge bundle; since $\mathrm{G}_{k}$ acts trivially on invariant differentials of elliptic curves $E\to S$ where $S$ is a $k$-scheme, the action of $\mathrm{G}_{k}$ on $\operatorname{Pic}(\mathscr{M}^{\mathrm{sep}})$ is trivial. Hence we have

[TABLE]

where equality 1 is by [Fu11, 4.3.7] and equality 2 is since $\mathrm{G}_{k}\simeq\widehat{\mathbb{Z}}$. We have

[TABLE]

where equality 1 is by the computation for an algebraically closed field (Theorem 1.1) and also the fact that $\mathrm{H}^{2}_{\mathrm{\acute{e}t}}(\mathscr{M}^{\mathrm{sep}},\mathbb{G}_{m})$ is a torsion group (see [AM16, Proposition 2.5 (iii)]) and equality 2 is because any group action on the group of order $2$ is necessarily trivial. Thus 5.9.1 reduces to a natural extension

[TABLE]

and it remains to see whether 5.9.2 is split. It suffices to compute the size of $(\operatorname{Br}\mathscr{M})[2]$, since $(\operatorname{Br}\mathscr{M})[2]$ has $4$ or $2$ elements depending on whether 5.9.2 is split or not, respectively.

As in 5.2, the fppf Kummer sequence

[TABLE]

gives an exact sequence

[TABLE]

of abelian groups. We compute $\mathrm{H}_{\operatorname{fppf}}^{2}(\mathscr{M},\mu_{2})$ using the Leray spectral sequence which is of the form

[TABLE]

with differentials $\mathrm{E}_{2}^{p,q}\to\mathrm{E}_{2}^{p+2,q-1}$. We have

[TABLE]

from the fppf Kummer sequence on $\mathscr{M}^{\mathrm{sep}}$, where the $p=0$ case follows since we are in characteristic $2$ and $\Gamma(\mathscr{M}^{\mathrm{sep}},\mathbb{G}_{m})=\Gamma(\mathbb{A}_{k^{\mathrm{sep}}}^{1},\mathbb{G}_{m})=(k^{\mathrm{sep}})^{\times}$, the $p=1$ case is since the multiplication-by-2 map on $\Gamma(\mathscr{M}^{\mathrm{sep}},\mathbb{G}_{m})=(k^{\mathrm{sep}})^{\times}$ is an isomorphism, and the $p=2$ case is by the computation in the algebraically closed case (combine 5.2.5, 5.4.5, 5.4.6, 5.4.14).

Since $k$ has characteristic 2, the 2-cohomological dimension of $k$ satisfies $\mathrm{cd}_{2}(k)\leq 1$ by e.g. [GS06, 6.1.9]; hence $\mathrm{E}_{2}^{p,q}=0$ for $p\geq 2$ and any $q$. Hence there is an exact sequence

[TABLE]

of abelian groups. As above, the $\mathrm{G}_{k}$-action on $\mathrm{H}^{1}_{\mathrm{fppf}}(\mathscr{M}^{\mathrm{sep}},\mu_{2})$ is necessarily trivial so we have an isomorphism $\mathrm{H}^{1}(\mathrm{G}_{k},\mathrm{H}^{1}_{\mathrm{fppf}}(\mathscr{M}^{\mathrm{sep}},\mu_{2}))\simeq\operatorname{Hom}_{\mathrm{cont}}(\mathrm{G}_{k},\mathbb{Z}/(2))\simeq\mathbb{Z}/(2)$.

To describe $\mathrm{H}^{0}(\mathrm{G}_{k},\mathrm{H}^{2}_{\mathrm{fppf}}(\mathscr{M}^{\mathrm{sep}},\mu_{2}))$, we describe the $\mathrm{G}_{k}$-action on $\mathrm{H}^{2}_{\mathrm{fppf}}(\mathscr{M}^{\mathrm{sep}},\mu_{2})$. Let

[TABLE]

be a fixed root of $x^{2}+x+1$ (i.e. a primitive 3rd root of unity).

If $\xi\in k$, then $\mathrm{G}_{k}$ acts trivially on $\mathrm{H}^{2}_{\mathrm{fppf}}(\mathscr{M}^{\mathrm{sep}},\mu_{2})$; hence $\mathrm{H}^{0}(\mathrm{G}_{k},\mathrm{H}^{2}_{\mathrm{fppf}}(\mathscr{M}^{\mathrm{sep}},\mu_{2}))$ has $4$ elements, hence $\mathrm{H}^{2}_{\mathrm{fppf}}(\mathscr{M},\mu_{2})$ has $8$ elements by 5.9.5, hence $(\operatorname{Br}\mathscr{M})[2]$ has $4$ elements by 5.9.4, hence $\operatorname{Br}\mathscr{M}\simeq\mathbb{Z}/(2)\oplus\mathbb{Z}/(12)$.

Suppose $\xi\not\in k$. The $k$-algebra map

[TABLE]

sending $\mu\mapsto(\nu_{1},\nu_{2})$ and $\omega\mapsto(\xi,\xi^{2})$ induces an isomorphism

[TABLE]

of $k^{\mathrm{sep}}$-algebras. The inverse to 5.9.6 sends

[TABLE]

for $f_{i}(\nu_{i})\in k[\nu_{i},\frac{1}{\nu_{i}^{3}-1}]$.

Let

[TABLE]

be an automorphism of $k^{\mathrm{sep}}$ such that $\lambda(\xi)=\xi^{2}$. Then the $k$-algebra automorphism of $k^{\mathrm{sep}}[\nu_{1},\frac{1}{\nu_{1}^{3}-1}]\times k^{\mathrm{sep}}[\nu_{2},\frac{1}{\nu_{2}^{3}-1}]$ induced by 5.9.6 sends $(\nu_{1},0)\mapsto(0,\nu_{2})$ and $(0,\nu_{2})\mapsto(\nu_{1},0)$ and $(\xi,0)\mapsto(0,\xi^{2})$ and $(0,\xi)\mapsto(\xi^{2},0)$. We see that the action of $\lambda$ on $M$ (see 5.3.2) is given by 5.9.7.

[TABLE]

A computation with 5.9.7 and 5.3.6 shows that

[TABLE]

for any $m\in M$ and $\mathsf{g}\in\operatorname{GL}_{2,3}$.

Let $f_{\operatorname{GL}_{2,3}}:\operatorname{GL}_{2,3}\to M$ be an inhomogeneous 1-cocycle as in Note 5.7. Multiplying the 1-cocycle condition 5.4.8 on the left by $\lambda$ gives

[TABLE]

where equality 1 follows from 5.9.8. Hence the function $\lambda\cdot f_{\operatorname{GL}_{2,3}}:\operatorname{GL}_{2,3}\to M$ sending $\mathsf{g}\mapsto\lambda\cdot f_{\operatorname{GL}_{2,3}}(\mathsf{g})$ is a 1-cocycle as well. Using 5.9.7 and 5.7.4, we have that

[TABLE]

and so

[TABLE]

for the same $s,s_{1}\in\mathbb{Z}/(2)$ as in 5.7.4.

Suppose $f_{\operatorname{GL}_{2,3}}$ and $\lambda\cdot f_{\operatorname{GL}_{2,3}}$ differ by a 1-coboundary, in other words there exists an element

[TABLE]

such that

[TABLE]

for all $\mathsf{g}\in\operatorname{GL}_{2,3}$. By 5.9.10, taking $\mathsf{g}=\mathsf{M}_{2}$ in 5.9.11 gives $m^{i}:=m_{1}^{i}=m_{2}^{i}=m_{3}^{i}$ for $i=1,2$; then taking $\mathsf{g}=\mathsf{M}_{1}$ gives $m^{1}=m^{2}$; then taking $\mathsf{g}=\mathsf{i}$ gives $m=0$. We see that $f_{\operatorname{GL}_{2,3}}$ and $\lambda\cdot f_{\operatorname{GL}_{2,3}}$ differ by a 1-coboundary if and only if $s=0$.

Hence we have that $\mathrm{H}^{0}(\mathrm{G}_{k},\mathrm{H}^{2}_{\mathrm{fppf}}(\mathscr{M}^{\mathrm{sep}},\mu_{2}))\simeq\mathbb{Z}/(2)$, hence $\mathrm{H}^{2}_{\mathrm{fppf}}(\mathscr{M},\mu_{2})$ has $4$ elements by 5.9.5, hence $(\operatorname{Br}\mathscr{M})[2]$ has $2$ elements by 5.9.4, hence $\operatorname{Br}\mathscr{M}\simeq\mathbb{Z}/(24)$. ∎

Appendix A The Weierstrass and Hesse presentations of $[\Gamma(3)]$

The purpose of this section is to prove Proposition A.4 below, which we could not find proved in the literature. For completeness of exposition, we first recall the definition of a full level $N$ structure on an elliptic curve $E/S$.

A.1 (Full level $N$ structure).

[KM85, Ch. 3] Let $N$ be a positive integer. We define $[\Gamma(N)]$ to be the category of pairs

[TABLE]

where

[TABLE]

is an elliptic curve and

[TABLE]

is a morphism of $S$-group schemes inducing an isomorphism $(\mathbb{Z}/(N))^{2}_{S}\simeq E[N]$. A morphism

[TABLE]

is a pair

[TABLE]

of morphisms of schemes such that the diagram

[TABLE]

commutes, where the morphism $\operatorname{id}\times\beta$ is the one induced by the identity on $(\mathbb{Z}/(3))^{2}_{\mathbb{Z}}$ and $\beta$, and such that $\alpha$ induces an isomorphism of $S_{1}$-group schemes $E_{1}\simeq S_{1}\times_{\beta,S_{2}}E_{2}$.

There is a functor

[TABLE]

sending $(E/S,\xi)\mapsto E/S$ on objects and $(\alpha,\beta)\mapsto(\alpha,\beta)$ on morphisms. If $E/S$ admits a full level $N$ structure, then $N$ is invertible on $S$ by [KM85, 2.3.2], hence the above functor factors through $\mathscr{M}_{1,1,\mathbb{Z}[\frac{1}{N}]}$. If $N\geq 3$, then for any scheme $S$ the fiber category $[\Gamma(N)](S)$ is equivalent to a set by [KM85, 2.7.2], so $[\Gamma(N)]$ is fibered in sets over the category of schemes.

A.2 (The $\operatorname{GL}_{2}(\mathbb{Z}/(N))$-action on $[\Gamma(N)]$).

Fix a scheme $S$. For any element

[TABLE]

in $\operatorname{GL}_{2}(\mathbb{Z}/(N))$, let

[TABLE]

be the $S$-group scheme automorphism of $(\mathbb{Z}/(N))^{2}_{S}$ corresponding to the abelian group homomorphism $(\mathbb{Z}/(N))^{2}\to(\mathbb{Z}/(N))^{2}$ defined by

[TABLE]

for $x_{1},x_{2}\in\mathbb{Z}/(N)$, i.e. acting by multiplication on the left on $(\mathbb{Z}/(N))^{2}$ viewed as vertical vectors. We have

[TABLE]

for $\sigma_{1},\sigma_{2}\in\operatorname{GL}_{2}(\mathbb{Z}/(N))$.

Fix an object $(E/S,\xi)\in[\Gamma(N)](E/S)$; then $(E/S,\xi\circ\varphi_{\sigma})$ is another object of $[\Gamma(N)](E/S)$, i.e. corresponds to another full level $N$ structure on $E/S$. This implies that there is a natural action of $\operatorname{GL}_{2}(\mathbb{Z}/(N))$ on each fiber category $[\Gamma(N)](E/S)$; the action is a right action since it is defined by precomposition.

Theorem A.3**.**

[KM85, 4.7.2]** If $N\geq 3$, the category $[\Gamma(N)]$ is representable by a smooth affine curve $Y(N)$ over $\mathbb{Z}[\frac{1}{N}]$.

We are primarily interested in the case $N=3$. The $3$-torsion points of an elliptic curve correspond to its inflection points (also “flex points”). In [KM85, (2.2.11)] it is shown that $Y(3)\simeq\operatorname{Spec}A_{\operatorname{W}}$ where

[TABLE]

and the universal elliptic curve over $A_{\operatorname{W}}$ with full level 3 structure is the pair

[TABLE]

where

[TABLE]

The formulas A.3.2 and A.3.3 are obtained by imposing the condition that the line $Y=X+BZ$ is a flex tangent to $E_{\operatorname{W}}$ at $[C:B+C:1]$. The ring $A_{\operatorname{W}}$ is isomorphic to $TMF(3)_{0}$ 5.6.1, with mutually inverse ring isomorphisms $TMF(3)_{0}\to A_{\operatorname{W}}$ and $A_{\operatorname{W}}\to TMF(3)_{0}$ given by $(\zeta,t)\mapsto(\frac{B+C}{C},\frac{1}{3C})$ and $(B,C)\mapsto(\frac{1}{3(\zeta-1)t},\frac{1}{3t})$ respectively.

In this paper, however, we use the “Hesse presentation” of $Y(3)$ as in [FO10, 5.1]. The following is claimed without proof in the Introduction to [DR73] and [Har11, 5.2.30].

Proposition A.4**.**

There is an isomorphism $Y(3)\simeq\operatorname{Spec}A_{\operatorname{H}}$ where

[TABLE]

and the universal elliptic curve over $A_{\operatorname{H}}$ with full level 3 structure is the pair

[TABLE]

with identity section $[1:-1:0]$.

The explicit $\mathbb{Z}[\frac{1}{3}]$-algebra isomorphisms $A_{\operatorname{H}}\to A_{\operatorname{W}}$ and $A_{\operatorname{W}}\to A_{\operatorname{H}}$ are given in A.8.7 and A.8.8 respectively.

A.5.

By [Sma01, §4], the group law of an elliptic curve $E=\operatorname{Proj}A[X,Y,Z]/(X^{3}+Y^{3}+Z^{3}=3\mu XYZ)$ in Hessian form over a ring $A$ is as follows. If $P=[x:y:z]$, then $2P=[x^{\prime}:y^{\prime}:z^{\prime}]$ where

[TABLE]

and if $P_{i}=[x_{i}:y_{i}:z_{i}]$ are points of $E_{\operatorname{H}}$ for $i=1,2,3$ satisfying $P_{1}+P_{2}=P_{3}$, then

[TABLE]

which only makes sense if $P_{1}\neq P_{2}$.

Using the above formulas, we may check that the full level 3 structure $\xi_{\operatorname{H}}:(\mathbb{Z}/(3))^{2}_{A_{\operatorname{H}}}\to E_{\operatorname{H}}$ is given by the table A.5.1.

[TABLE]

The Hesse presentation A.4.1 is sometimes easier to work with than the Weierstrass presentation A.3.1 since the equation of the universal elliptic curve is symmetric in $X,Y,Z$, which means that there is also considerable symmetry in the 3-torsion points A.5.1.

A.6.

We describe the $\operatorname{GL}_{2}(\mathbb{Z}/(3))$-action on $E_{\operatorname{H}}/A_{\operatorname{H}}$. Set $S_{\mathrm{H}}:=\operatorname{Spec}A_{\mathrm{H}}$. The functor $[\Gamma(3)]$ being representable by $S_{\mathrm{H}}$ means explicitly that for any $\mathbb{Z}[\frac{1}{3}]$-scheme $T$ and object $(E/T,\xi)\in([\Gamma(3)])(T)$, there exists a unique pair $(\alpha,\beta)$ of morphisms of schemes $\alpha:E\to E_{\mathrm{H}}$ and $\beta:T\to S_{\mathrm{H}}$ such that the diagram

{E}$${E_{\mathrm{H}}}$${(\mathbb{Z}/(3))^{2}_{T}}$${(\mathbb{Z}/(3))^{2}_{S_{\mathrm{H}}}}$${T}$${S_{\mathrm{H}}}$$\alpha$$\beta$$\operatorname{id}\times\beta$$\xi$$\xi_{\mathrm{H}}$$f_{T}$$f_{S_{\mathrm{H}}}

commutes and induces an isomorphism of $T$-group schemes $E\simeq T\times_{\beta,S_{\mathrm{H}}}E_{\mathrm{H}}$ as in A.1.1.

As in A.2, for every $\sigma\in\operatorname{GL}_{2}(\mathbb{Z}/(3))$, let $\varphi_{\sigma}$ be the $S_{\mathrm{H}}$-automorphism of $(\mathbb{Z}/(3))^{2}_{S_{\mathrm{H}}}$ induced by $\sigma$; then precomposition $\xi_{\mathrm{H}}\varphi_{\sigma}$ defines another full level 3 structure on $E_{\mathrm{H}}/S_{\mathrm{H}}$. Taking $T=S_{\mathrm{H}}$ and $\xi=\xi_{\mathrm{H}}\varphi_{\sigma}$ above, there is a unique pair $(\alpha_{\sigma},\beta_{\sigma})$ of morphisms of schemes $\alpha_{\sigma}:E_{\mathrm{H}}\to E_{\mathrm{H}}$ and $\beta_{\sigma}:S_{\mathrm{H}}\to S_{\mathrm{H}}$ such that the diagram

{E_{\mathrm{H}}}$${E_{\mathrm{H}}}$${(\mathbb{Z}/(3))^{2}_{S_{\mathrm{H}}}}$${(\mathbb{Z}/(3))^{2}_{S_{\mathrm{H}}}}$${S_{\mathrm{H}}}$${S_{\mathrm{H}}}$$\alpha_{\sigma}$$\beta_{\sigma}$$\operatorname{id}\times\beta_{\sigma}$$\xi_{\mathrm{H}}\varphi_{\sigma}$$\xi_{\mathrm{H}}$$f_{S_{\mathrm{H}}}$$f_{S_{\mathrm{H}}}

commutes and induces an isomorphism of $S_{\mathrm{H}}$-group schemes $E_{\mathrm{H}}\simeq S_{\mathrm{H}}\times_{\beta_{\sigma},S_{\mathrm{H}}}E_{\mathrm{H}}$. Given two elements $\sigma_{1},\sigma_{2}\in\operatorname{GL}_{2}(\mathbb{Z}/(3))$, we have a commutative diagram

{E_{\mathrm{H}}}$${E_{\mathrm{H}}}$${E_{\mathrm{H}}}$${(\mathbb{Z}/(3))^{2}_{S_{\mathrm{H}}}}$${(\mathbb{Z}/(3))^{2}_{S_{\mathrm{H}}}}$${(\mathbb{Z}/(3))^{2}_{S_{\mathrm{H}}}}$${S_{\mathrm{H}}}$${S_{\mathrm{H}}}$${S_{\mathrm{H}}}$$\alpha_{\sigma_{1}}$$\alpha_{\sigma_{2}}$$\beta_{\sigma_{1}}$$\beta_{\sigma_{2}}$$\xi_{\mathrm{H}}\varphi_{\sigma_{1}}\varphi_{\sigma_{2}}$$\xi_{\mathrm{H}}\varphi_{\sigma_{2}}$$\xi_{\mathrm{H}}$$f_{S_{\mathrm{H}}}$$f_{S_{\mathrm{H}}}$$f_{S_{\mathrm{H}}}

which implies

[TABLE]

since $\varphi_{\sigma_{1}\sigma_{2}}=\varphi_{\sigma_{1}}\varphi_{\sigma_{2}}$ (see A.2). Thus the assignment

[TABLE]

defines a right action of $\operatorname{GL}_{2}(\mathbb{Z}/(3))$ on the scheme $S_{\mathrm{H}}$.

In terms of the generators

[TABLE]

of $\operatorname{GL}_{2}(\mathbb{Z}/(3))$, the action of $\operatorname{GL}_{2}(\mathbb{Z}/(3))$ on $E_{\mathrm{H}}/A_{\mathrm{H}}$ is as follows. (We refer to A.5.1 for the additive structure on $E_{\operatorname{H}}[3]$.)

1. (1)

For $\sigma=\mathsf{M}_{1}$, the new level 3 structure $\xi_{\mathrm{H}}\varphi_{\mathsf{M}_{1}}$ is

[TABLE]

and the scheme morphisms $\alpha_{\mathsf{M}_{1}}:E_{\operatorname{H}}\to E_{\operatorname{H}}$ and $\beta_{\mathsf{M}_{1}}:S_{\mathrm{H}}\to S_{\mathrm{H}}$ correspond to the ring homomorphisms sending

[TABLE]

respectively. 2. (2)

For $\sigma=\mathsf{M}_{2}$, the new level 3 structure $\xi_{\mathrm{H}}\varphi_{\mathsf{M}_{2}}$ is

[TABLE]

and the scheme morphisms $\alpha_{\mathsf{M}_{2}}:E_{\operatorname{H}}\to E_{\operatorname{H}}$ and $\beta_{\mathsf{M}_{2}}:S_{\mathrm{H}}\to S_{\mathrm{H}}$ correspond to the ring homomorphisms sending

[TABLE]

respectively. 3. (3)

For $\sigma=\mathsf{i}$, the new level 3 structure $\xi_{\mathrm{H}}\varphi_{\mathsf{i}}$ is

[TABLE]

and the scheme morphisms $\alpha_{\mathsf{i}}:E_{\operatorname{H}}\to E_{\operatorname{H}}$ and $\beta_{\mathsf{i}}:S_{\mathrm{H}}\to S_{\mathrm{H}}$ correspond to the ring homomorphisms sending

[TABLE]

respectively.

Remark A.7.

According to our convention, the action of $\operatorname{GL}_{2}(\mathbb{Z}/(3))$ on the fiber category $[\Gamma(3)](E_{\operatorname{H}}/\operatorname{Spec}A_{\operatorname{H}})$ is by precomposition, hence the action of $\operatorname{GL}_{2}(\mathbb{Z}/(3))$ on pairs of points on the right hand side of A.5.1 is a right action; thus the induced action of $\operatorname{GL}_{2}(\mathbb{Z}/(3))$ on the scheme $\operatorname{Spec}A_{\operatorname{H}}$ is a right action (as described in A.6.1) and the corresponding action of $\operatorname{GL}_{2}(\mathbb{Z}/(3))$ on the coordinate ring $A_{\operatorname{H}}$ is a left action.

A.8 (Proof of Proposition A.4).

In fact, it turns out that the identities

[TABLE]

hold in $A_{\operatorname{W}}$ which yields a simpler description

[TABLE]

of $A_{\operatorname{W}}$. (For A.8.1, write out $a_{1}^{3}-27a_{3}$ in terms of $B,C$ and notice that it is of the form $9C+27B-1$ plus higher order terms; then check that the naive guess works. To see A.8.2, substitute $C^{2}=-3CB-3B^{2}$ into A.3.3.)

We follow the argument of [AD09, 2.1]; see also [Con96, §1.4.1, §1.4.2]. Working “generically”, we will assume that $a_{1}$ is a unit to obtain the coordinate change formula A.8.9, then observe that it applies also to the case when $a_{1}$ is not a unit. Starting with

[TABLE]

we define $X_{2},Y_{2},Z_{2}$ by the system

[TABLE]

where $u=a_{1}/3$ and substitute into A.8.3 to get

[TABLE]

We define $X_{3},Y_{3},Z_{3}$ by the system

[TABLE]

where $\omega=\frac{C+B}{B}$ 222Since $3$ is invertible, if $x$ is a root of the polynomial $T^{2}+3T+3$ then $x+1$ is a root of the polynomial $T^{2}+T+1$, thus it is natural to take $\frac{C+B}{B}$ as our $\omega$. and substitute into A.8.4 to get

[TABLE]

or equivalently

[TABLE]

We know that the coefficient of $Z_{3}^{3}$ in A.8.5 is a cube A.8.1 so we normalize by defining $X_{4},Y_{4},Z_{4}$ by the system

[TABLE]

and substitute into A.8.5 to get

[TABLE]

To summarize the above, there is a ring homomorphism $\varphi_{21}:A_{\operatorname{H}}\to A_{\operatorname{W}}$ sending

[TABLE]

and solving for $B,C$ in terms of $\mu,\omega$ implies that the inverse $\varphi_{12}:A_{\operatorname{W}}\to A_{\operatorname{H}}$ sends

[TABLE]

where $\omega+2$ is a unit of $A_{\operatorname{H}}$ since $(\omega+2)(\omega-1)=-3$ and $\mu-\omega$ is a unit of $A_{\operatorname{H}}$ since $\mu^{3}-1=(\mu-1)(\mu-\omega)(\mu-\omega^{2})$. We may check that the product

[TABLE]

is “projectively equivalent” to the matrix

[TABLE]

whose determinant is a unit of $A_{\operatorname{W}}$. Given a section $[s_{X}:s_{Y}:s_{Z}]$ of A.8.3, the corresponding section of A.8.6 is $X^{-1}\cdot[s_{X}:s_{Y}:s_{Z}]^{\operatorname{T}}$ where

[TABLE]

The above implies that the sections

[TABLE]

of A.8.3 (i.e. the identity section and ordered basis for the 3-torsion) correspond to the sections

[TABLE]

of A.8.6. We may apply an automorphism of the pair $(A_{\operatorname{H}},E_{\operatorname{H}}/A_{\operatorname{H}})\in\mathscr{M}_{1,1,\mathbb{Z}}$ of the form A.6(2) (for $Y$ instead of $Z$) to A.8.10 to get

[TABLE]

and using the fact that there is a simply transitive action of $\operatorname{GL}_{2}(\mathbb{Z}/(3))$ on the set of ordered bases of the 3-torsion in $E_{\operatorname{H}}/A_{\operatorname{H}}$, we may switch the second and third sections of A.8.11 to obtain

[TABLE]

as desired. ∎

Remark A.9.

For A.8.1, see also Stojanoska’s derivation [Sto14, §4.1].

Remark A.10.

There are coordinate change formulas in [Sma01, §3] transforming a Weierstrass equation into Hesse normal form, but there it is assumed that the base ring is a finite field $\mathbb{F}_{q}$ where $q\equiv 2\pmod{3}$, in order to take cube roots of $a_{1}^{3}-27a_{3}$, but from this description it is not clear that the cube root is an algebraic function. As shown in A.8.1, it turns out that in fact $a_{1}^{3}-27a_{3}$ is a cube in the ring $A_{\operatorname{W}}$. One suspects that this is the case after tracing through the proof of [AD09, 2.1] and arriving at the equation $x^{3}+y^{3}+\frac{27a_{3}-a_{1}^{3}}{a_{1}^{3}}z^{3}=3xyz$, in which case we know that $\frac{27a_{3}-a_{1}^{3}}{a_{1}^{3}}$ is a cube by Lemma A.11.

Lemma A.11**.**

Let $k$ be a field of characteristic not $3$, and let

[TABLE]

be a curve in $\mathbb{A}_{k}^{2}$. Suppose that

[TABLE]

is the tangent line to a flex point of $E$ and suppose that $a^{3}\neq b^{3}$. Then $\beta$ is a cube in $k$.

Proof.

If $a=0$, then $b\neq 0$ and substituting $y=-\frac{c}{b}$ into A.11.1 and rearranging gives $x^{3}+\frac{3c}{b}x-(\frac{c}{b})^{3}+\beta=0$ which by assumption is of the form $(x+\ell)^{3}$ for some $\ell\in k$. Comparing coefficients, we have $\ell=0$ and so $\beta=(\frac{c}{b})^{3}$.

By symmetry we may assume that $a,b\neq 0$. By scaling A.11.2, we may assume that $b=-1$. Substituting $y=ax+c$ into $E$ gives

[TABLE]

and dividing by the leading coefficient gives

[TABLE]

and comparing this to

[TABLE]

gives either $ac-1=0$ in which case $c^{3}+\beta=0$ as well (so that $\beta=(-1/a)^{3}=(-c)^{3}$), otherwise if $ac-1\neq 0$ then

[TABLE]

which implies $c=-a^{2}$ so that the original equation of the tangent line is $y=ax-a^{2}$. Substituting this back into $E$ gives $\beta=(-a)^{3}$. ∎

Appendix B Higher direct images of sheaves on classifying stacks of discrete groups

The material in this section is standard and we claim no originality.

For a category $\mathcal{C}$, we denote by $\operatorname{PSh}(\mathcal{C})$ (resp. $\operatorname{PAb}(\mathcal{C})$) the category of presheaves (resp. abelian presheaves) on $\mathcal{C}$. If $\mathcal{C}$ is a site, we denote by $\operatorname{Sh}(\mathcal{C})$ (resp. $\operatorname{Ab}(\mathcal{C})$) the category of sheaves (resp. abelian sheaves) on $\mathcal{C}$.

Let $\mathcal{C}$ be a site, let $G$ be a finite (discrete) group, let $\mathrm{B}G_{\mathcal{C}}$ be the classifying stack associated to $G$ over $\mathcal{C}$. Let

[TABLE]

be the projection and let

[TABLE]

be the canonical section of $\pi$. We view any fibered category $p:\mathcal{F}\to\mathcal{C}$ as a site via the Grothendieck topology inherited from $\mathcal{C}$ via $p$.

Lemma B.1**.**

In the setup above, for any abelian sheaf $\mathscr{F}\in\operatorname{Ab}(\mathrm{B}G_{\mathcal{C}})$ the higher pushforward $\mathbf{R}^{i}\pi_{\ast}\mathscr{F}$ is naturally isomorphic to the sheaf associated to the presheaf whose value on an object $U\in\mathcal{C}$ is $\mathrm{H}^{i}(G,\Gamma(U,\varphi^{\ast}\mathscr{F}))$.

Proof.

Let $\mathrm{P}G_{\mathcal{C}}$ denote the category whose objects are the objects of $\mathcal{C}$ and where a morphism $X_{1}\to X_{2}$ in $\mathrm{P}G_{\mathcal{C}}$ is a pair $(\varphi,g)$ where $\varphi\in\operatorname{Mor}_{\mathcal{C}}(X_{1},X_{2})$ and $g\in G$. (In other words, there is an equivalence of categories $\mathrm{P}G_{\mathcal{C}}\simeq\mathcal{C}\times[\ast/G]$ where $[\ast/G]$ is the category with one object $\ast$ and where $\operatorname{Hom}_{[\ast/G]}(\ast,\ast)$ is isomorphic to $G$.) The fibered category $\mathrm{P}G_{\mathcal{C}}$ is a (separated) prestack whose associated stack is $\mathrm{B}G_{\mathcal{C}}$, and the inclusion $\mathrm{P}G_{\mathcal{C}}\to\mathrm{B}G_{\mathcal{C}}$ induces an equivalence of topoi $\operatorname{Sh}(\mathrm{P}G_{\mathcal{C}})\simeq\operatorname{Sh}(\mathrm{B}G_{\mathcal{C}})$. Hence in the statement of the lemma we may replace $\mathrm{B}G_{\mathcal{C}}$ by $\mathrm{P}G_{\mathcal{C}}$ where by abuse of notation we also denote

[TABLE]

the projection morphism. Since sheafification is an exact functor, the diagram

{\operatorname{PAb}(\mathrm{P}G_{\mathcal{C}})}$${\operatorname{PAb}(\mathcal{C})}$${\operatorname{Ab}(\mathrm{P}G_{\mathcal{C}})}$${\operatorname{Ab}(\mathcal{C})}$$\pi^{\operatorname{pre}}_{\ast}$$\pi_{\ast}$$\operatorname{sh}$$\operatorname{sh}

is (2-)commutative. For the same reason, we have a natural isomorphism

[TABLE]

in $\operatorname{D}^{+}(\operatorname{Ab}(\mathcal{C}))$ for any abelian presheaf $\mathscr{F}\in\operatorname{PAb}(\mathrm{P}G_{\mathcal{C}})$. Presheaves on $\mathrm{P}G_{\mathcal{C}}$ correspond to presheaves $\mathscr{F}$ on $\mathcal{C}$ equipped with a $G$-action, and under this identification $\pi^{\operatorname{pre}}_{\ast}(\mathscr{F})=\mathscr{F}^{G}$ where $\Gamma(U,\mathscr{F}^{G}):=(\Gamma(U,\mathscr{F}))^{G}$ for all $U\in\mathcal{C}$. Let $\mathscr{F}\in\operatorname{Ab}(\mathrm{P}G_{\mathcal{C}})$ be an abelian sheaf, and let

[TABLE]

be a resolution of $\mathscr{F}$ by injective abelian presheaves $\mathcal{I}^{i}\in\operatorname{PAb}(\mathrm{P}G_{\mathcal{C}})$. Then $\mathbf{R}\pi^{\operatorname{pre}}_{\ast}(\mathscr{F})$ is isomorphic to

[TABLE]

in $\operatorname{D}^{+}(\operatorname{PAb}(\mathcal{C}))$, and $\Gamma(U,\mathbf{R}\pi^{\operatorname{pre}}_{\ast}(\mathscr{F}))$ is isomorphic to

[TABLE]

in $\operatorname{D}^{+}(\operatorname{PAb}(\mathcal{C}))$. Furthermore $\Gamma(U,\mathcal{I}^{i})\simeq(i_{U})^{\ast}\mathcal{I}^{i}$ is an injective $G$-module for all $i$ by Lemma B.2, thus we have an isomorphism

[TABLE]

of abelian groups. ∎

Lemma B.2**.**

Let $\mathcal{C}$ be a category, let $U\in\mathcal{C}$ be an object, let $\mathcal{A}_{\mathcal{C},U}$ denote the full subcategory of $\mathcal{C}$ containing exactly $U$, and let $i_{U}:\mathcal{A}_{\mathcal{C},U}\to\mathcal{C}$ denote the inclusion. The inverse image functor $(i_{U})^{\ast}:\operatorname{PAb}(\mathcal{C})\to\operatorname{PAb}(\mathcal{A}_{\mathcal{C},U})$ preserves injectives.

Proof.

The functor $(i_{U})^{\ast}:\operatorname{PAb}(\mathrm{P}G_{\mathcal{C}})\to\operatorname{PAb}(\mathcal{A}_{\mathcal{C},U})$ has an exact left adjoint, namely the “extension by zero” functor $i_{U,!}:\operatorname{PAb}(\mathcal{A}_{\mathcal{C},U})\to\operatorname{PAb}(\mathrm{P}G_{\mathcal{C}})$ which sends $M\in\operatorname{PAb}(\mathcal{A}_{\mathcal{C},U})$ to the abelian presheaf $i_{U,\dagger}(M)$ where $\Gamma(V,i_{U,\dagger}(M))=M$ if $V=U$ and [math] otherwise (with the only nontrivial restriction morphisms being those corresponding to the endomorphisms of $U$). ∎

Appendix C Computation using Magma

We compute $\mathrm{H}^{1}(\operatorname{GL}_{2}(\mathbb{Z}/(3)),M)$ in 5.4 using Magma [BCP97]. Here G is defined as the subgroup of $\operatorname{GL}_{2}(\mathbb{Z}/(3))$ generated by the matrices in 5.3.6, but the specified matrices constitute a generating set so in fact G $=\operatorname{GL}_{2}(\mathbb{Z}/(3))$. The group G acts on the abelian group $M=(\mathbb{Z}/(2))^{\oplus 6}$ by the three specified elements of $\operatorname{Mat}_{6\times 6}(\mathbb{Z})$, where each $\mathbf{x}\in M$ is viewed as a horizontal vector and each $6\times 6$ matrix $\mathbf{A}$ acts on $M$ by right multiplication $\mathbf{x}\mapsto\mathbf{x}\cdot\mathbf{A}$. The last line computes $\mathrm{H}^{1}(G,(\mathbb{Z}/(2))^{\oplus 6})$.

G := MatrixGroup< 2 , FiniteField(3) | [ 1,0 , -1,1 ] , [ 0,-1 , 1,0] , [ 1,0 , 0,-1 ]

; mats := [ Matrix(Integers() , 6 , 6 , [ 0, 0, 1, 0, 0, 0 , 1, 0, 0, 0, 0, 0 , 0, 1, 0, 0, 0, 0 , 0, 0, 0, 0, 1, 0 , 0, 0, 0, 0, 0, 1 , 0, 0, 0, 1, 0, 0 ]) , Matrix(Integers() , 6 , 6 , [ 1, 0, 0, 0, 0, 0 , 1, 0, 1, 0, 0, 0 , 1, 1, 0, 0, 0, 0 , 0, 0, 0, 1, 0, 0 , 0, 0, 0, 1, 0, 1 , 0, 0, 0, 1, 1, 0 ]) , Matrix(Integers() , 6 , 6 , [ 0, 0, 0, 1, 0, 0 , 0, 0, 0, 0, 1, 0 , 0, 0, 0, 0, 0, 1 , 1, 0, 0, 0, 0, 0 , 0, 1, 0, 0, 0, 0 , 0, 0, 1, 0, 0, 0 ]) ]; CM := CohomologyModule(G,[2,2,2,2,2,2],mats); CohomologyGroup(CM,1);