2.1 Known rationality results

The coarse moduli spaces ${\mathcal{A}}_g$ have been much studied over the complex numbers from a birational point of view. With the exception of the case ( $g=6$ ), the broad picture remains as described in [Reference Hulek and SankaranHS02], to which we refer for more details.

2.2 Moduli of curves

In view of Katsylo’s result, it is natural to ask whether ${\mathcal{A}}_3$ is rational over $\mathbb{Q}$ . What Katsylo proves directly, however, is that $\mathcal{M}_3$ is rational (over $\mathbb{C}$ ), and the wider context into which the proof naturally fits is the moduli of curves rather than of abelian varieties. Katsylo’s proof is one of many rationality proofs for moduli spaces of curves: for some examples, see [Reference Verra, Farkas and MorrisonVer13]. The main tool for many of these is classical invariant theory. Few of them are written with much attention to fields of definition. Shepherd-Barron’s proof [Reference Shepherd-BarronS-B89] that $\mathcal{M}_6$ is rational over $\mathbb{Q}$ is an exception. In fact, Katsylo’s proof for $\mathcal{M}_3$ is among the most complicated of these arguments.

2.3 Prym curves

A Prym curve (of genus $g$ ) over a scheme $S$ is a pair $(D,C)$ where $C$ is a smooth proper scheme over $S$ whose fibres are smooth projective geometrically integral curves of genus $g$ , and $D \to C$ is a finite étale morphism of degree $2$ whose fibres over $S$ are geometrically integral. We denote by $\mathscr{R}_g$ the moduli stack of Prym curves of genus $g$ and by ${\mathcal{R}}_g$ its coarse moduli space.

An explicit construction of this stack over ${\mathbb{Z}}[1/2]$ can be found in the proof of [Reference BeauvilleBea77, Theorem 6.5], which also constructs a morphism (the Prym map) $\mathscr{R}_g \to \mathscr{A}_{g-1}$ of stacks over ${\mathbb{Z}}[1/2]$ . Moreover, $\mathscr{R}_g \to \mathscr{A}_{g-1}$ is dominant for $g \leq 6$ , by [Reference BeauvilleBea77, Lemma 6.5.2].

3.1 Stable rationality of moduli of hypersurfaces

We will prove that ${\mathcal{A}}_3$ is stably rational by showing that the moduli space of plane quartic curves is stably rational. Our argument is sufficiently robust that it also works for other moduli of hypersurfaces. Our result is as follows.

Proof. The case $d = 2$ is classical so we assume $d \gt 2$ . Let $V_{d,n}={{\rm H}}^0({\mathbb{P}}^n,{\mathcal{O}}_{{\mathbb{P}}^n}(d))$ be the vector space of forms of degree $d$ in $(n+1)$ variables. The group ${\rm SL}_{n+1}$ acts on $V_{d,n}$ in a natural way via linear change of variables. The coprimality conditions ensure that the central copy of $\mu _{n+1} \subset {\rm SL}_{n+1}$ acts faithfully on $V_{d,n}$ . Moreover, the induced action of ${\rm SL}_{n+1}$ on $\mathcal{H}_{d,n}$ factors through the ${\rm SL}_{n+1}/\mu _{n+1}={\rm PGL}_{n+1}$ -action. We obtain the following commutative diagram.

The action of ${\rm SL}_{n+1}$ on $V_{d,n}$ is generically free: indeed, $\mu _{n+1}$ acts faithfully and the action of ${\rm PGL}_{n+1}$ on $\mathcal{H}_{d,n}$ is generically free as there exist hypersurfaces of degree $d$ with trivial linear automorphism group [Reference PoonenPoo05]. We conclude that the generic fibre of the left-hand map is an ${\rm SL}_{n+1}$ -torsor. But such a torsor is necessary trivial as ${{\rm H}}^1(k,{\rm SL}_{n+1}) = 0$ for any field $k$ [Reference Platonov and RapinchukPR94, Lemma 2.3]. We conclude that $V_{d,n}$ is birational to ${\rm SL}_{n+1} \times (V_{d,n} /{\rm SL}_{n+1})$ .

However, ${\rm SL}_{n+1}$ is rational over $k$ . This can be seen by noting that the determinant is linear in the coordinate $x_{1,1}$ , and thus one can solve for $x_{1,1}$ . Thus $V_{d,n} /{\rm SL}_{n+1}$ is stably rational.

Next, the top arrow in the diagram is a ${\mathbb{G}}_{{\rm m}}$ -torsor. The bottom arrow is thus a ${\mathbb{G}}_{{\rm m}}/\mu _{n+1} ={\mathbb{G}}_{{\rm m}}$ -torsor (here the map $\mu _{n+1} \to{\mathbb{G}}_{{\rm m}}$ is the one induced by the ${\rm SL}_{n+1}$ -action and not necessarily the standard embedding). By Hilbert’s Theorem 90 the generic fibre is the trivial torsor: thus we see that $\mathcal{H}_{d,n}/{\rm PGL}_{n+1}$ is stably birational to $V_{d,n}/{\rm SL}_{n+1}$ , which we already proved is stably rational.

3.2 Abelian $3$ -folds

Let $k$ be a field of characteristic not equal to $3$ .

The rational maps $\mathcal{H}_{4,2}/{\rm PGL}_3 \dashrightarrow \mathcal{M}_3 \dashrightarrow{\mathcal{A}}_3$ over $k$ are birational over $\overline{k}$ (this follows from Torelli and a dimension count) and hence birational over $k$ . Therefore, ${\mathcal{A}}_3$ is stably rational over $k$ by Theorem3.1. This is sufficient for Theorem1.5.

3.3 Abelian 4-folds

Let $k$ be a field of characteristic $0$ . We prove the $g=4$ case of Theorem1.3.

For the following result we closely follow [Reference Izadi, Lo Giudice and SankaranILS09], which proves the analogous result over $\mathbb{C}$ , but a few modifications are required. The main difference is that in [Reference Izadi, Lo Giudice and SankaranILS09], the authors work with certain quartic surfaces, and then pass to the double covers of ${\mathbb{P}}^3$ ramified along the quartic surface. Some care is needed over non-algebraically closed fields where there may be many such covers given by quadratic twists.

In the rest of this section, we prove the first part of this: the second part follows since the Pyrm map ${\mathcal{R}}_5 \to{\mathcal{A}}_4$ is dominant and defined over $k$ .

Fix five points $P_1,\dots, P_5 \in{\mathbb{P}}^3(k)$ in linear general position. Without loss of generality, $P_1 =(0:0:0:1)$ . We consider the space $\mathcal{Q}'\subset{{\rm H}}^0({\mathcal{O}}_{{\mathbb{P}}^3}(4))$ of quartic forms $F$ in four variables such that the associated quartic surface $X:=\{x\in{\mathbb{P}}^3\mid F(x) = 0\}$ has exactly six ordinary double points over the algebraic closure, at least five of which are $P_1,\dots, P_5$ . This is a quasi-affine variety defined over $k$ . Moreover, if $F$ is defined over $k$ then necessarily the sixth double point is also defined over $k$ . The key result is as follows.

Proof. We follow [Reference Izadi, Lo Giudice and SankaranILS09, Proposition 2.1], which proves unirationality over the algebraic closure (our $\mathcal{Q}'$ is the affine cone over the $\mathcal{Q}$ appearing in [ILS]). The space of quartic polynomials $F$ whose associated quartic surface has at least five double points at $P_1,\dots, P_5$ is a vector space which we denote by $V$ . For the sixth double point, consider the scheme

\begin{equation*} B_0:= \{ (F, P_0) \in V \times {\mathbb {P}}^3 \mid F(P_0) = \partial F/\partial x_0 (P_0) = \dots = \partial F/\partial x_3(P_0) = 0\}. \end{equation*}

The projection onto ${\mathbb{P}}^3$ is surjective and the fibres are vector spaces (not just affine spaces: there is a section, the zero section). It follows that there is a dense open $U\subset{\mathbb{P}}^3$ over which $B_0$ becomes isomorphic to $U \times{\mathbb{A}}^n$ for some $n$ , and therefore $B_0$ is rational.

The other projection is a map $B_0 \to V$ whose image contains $\mathcal{Q}'$ as a dense open subset. As $B_0$ is rational, we see that $\mathcal{Q}'$ is unirational.

There is a natural map $ \varrho \colon \mathcal{Q}' \longrightarrow{\mathcal{R}}_5$ defined in the following way [ILS, Section 1]. For $F\in \mathcal{Q}'$ , let $X$ be the associated quartic surface and

\begin{equation*} \Lambda _X:=\{y^2 = F(x)\} \subset {\mathbb {P}}(1,1,1,1,2), \end{equation*}

the associated quartic double solid. Let $W_X$ be the blow-up of $\Lambda _X$ at $(0:0:0:1:0)$ . Then composing the blow-up map with the natural projection to ${\mathbb{P}}^2$ yields a morphism $f\colon W_X \to{\mathbb{P}}^2$ . This is a conic bundle morphism whose non-smooth locus $C_X \subset{\mathbb{P}}^2$ is a plane sextic curve whose singular points are exactly the images of the singular points of $X$ (see [ILS, Propositions 1.5 and 1.6]). Thus if $F$ is general, $C_X$ has five ordinary double points, so its normalisation $\widetilde{C}_X$ has genus $5$ .

Next let $S= W_X \times _{{\mathbb{P}}^2} \widetilde{C}_X$ and let $\widetilde{S}$ be the normalisation of $S$ . We then apply Stein factorisation to the induced map $\widetilde{S} \to \widetilde{C}_X$ to obtain a finite morphism $\Gamma _F \to \widetilde{C}_X$ , which by [ILS, Proposition 1.8] is a geometrically connected finite étale cover of degree $2$ . Thus the pair $(\widetilde{C}_X, \Gamma _F)$ gives a well-defined element of ${\mathcal{R}}_5$ . All these constructions are natural, so we define the morphism $\varrho$ by $\varrho (F)=(\tilde{C}_X,\Gamma _F)$ .

To show that ${\mathcal{R}}_5$ is unirational, we note from [ILS, Corollary 3.3] that $\mathcal{Q}'\to{\mathcal{R}}_5$ is dominant over $\overline{k}$ , hence it is dominant over $k$ , whence ${\mathcal{R}}_5$ is unirational by Lemma 3.3. As ${\mathcal{R}}_5 \to{\mathcal{A}}_4$ is dominant, ${\mathcal{A}}_4$ is also unirational.

3.4 Abelian 5-folds

Let $k$ be a field of characteristic $0$ . We prove the $g=5$ case of Theorem1.3.

We proceed as in Section 3.3, this time following [Reference Farkas and VerraFV16, Section 1], which proves the analogous result over $\mathbb{C}$ . This time the method extends to $k$ with little difficulty.

Pick four points $O_1,\dots, O_4 \in{\mathbb{P}}^2(k)$ in linear general position and let $P_i = (O_i,O_i) \in{\mathbb{P}}^2 \times{\mathbb{P}}^2$ . Consider the linear system ${\mathbb{P}}^{15}$ of hypersurfaces of bidegree $(2,2)$ in ${\mathbb{P}}^2 \times{\mathbb{P}}^2$ that have ordinary double points at $P_1,\,P_2,\,P_3,\,P_4$ . For such a 3-fold $X$ , projecting onto the first factor ${\mathbb{P}}^2$ induces a conic bundle morphism $X \to{\mathbb{P}}^2$ . The discriminant is a sextic curve $C_X$ whose singularities are generically the image of the singularities of $X$ [Reference Farkas and VerraFV16, Proposition 1.2]. Thus for general $X$ there are four nodes, so the normalisation $\widetilde{C}_X$ has genus $6$ . Moreover, applying Stein factorisation to the singular locus as in Section 3.3, we obtain a geometrically connected étale double cover $\Gamma _X \to \widetilde{C}_X$ . This constructs a rational map ${\mathbb{P}}^{15} \dashrightarrow{\mathcal{R}}_6$ over $k$ . By [Reference Farkas and VerraFV16, Theorem 1.4] this map is dominant, and hence ${\mathcal{R}}_6$ is unirational. Again ${\mathcal{R}}_6 \to{\mathcal{A}}_5$ is dominant, and thus ${\mathcal{A}}_5$ is also unirational.

Combining the results of this section with the results from Section 2.1 completes the proof of Theorem1.3.

4.1 $k_v$ -points of a stack

Let $k$ be a number field and $v$ a place of $k$ . For finitely presented algebraic stacks $\mathscr{X}$ over $k_v$ , Christensen [Reference ChristensenChr05, Section 5] gave a topology on $\mathscr{X}{({k_v})}$ , extending the $v$ -adic topology on the usual $k_v$ -points of schemes. It has the following properties.

1. (i) Any morphism of stacks over $k_v$ induces a continuous map on $k_v$ -points [Reference ChristensenChr05, Theorem 9.0.3].

2. (ii) Any smooth morphism of stacks over $k_v$ induces an open map on $k_v$ -points [Chr, Theorem 11.0.4].

The topology is unique because Christensen proves in [Chr, Theorem 7.0.7] that any $k_v$ -point of an algebraic stack is the image of a $k_v$ -point under a smooth morphism from some scheme.

The following corresponds to the well-known fact that a non-empty $v$ -adic open subset of a smooth irreducible scheme is Zariski dense, which is an application of the implicit function theorem.

We require the following version of Hensel’s lemma for algebraic stacks.

Lemma 4.2. Let $R$ be a complete noetherian local ring, or an excellent Henselian discrete valuation ring, with maximal ideal $\mathfrak{m}$ . Let $\mathscr{X}$ be a smooth algebraic stack over $R$ . Then for all $n \in{\mathbb{N}}$ the natural map

\begin{equation*} \mathscr {X}(R) \longrightarrow \mathscr {X}(R/{\mathfrak {m}}^n), \end{equation*}

is surjective.

Proof. Recall that smooth means formally smooth and locally of finite presentation. Formal smoothness (see [SP23, Tag 0DNV]) implies that for any $n \in{\mathbb{N}}$ and any $1$ -commutative diagram

there exists a diagonal arrow making the diagram $2$ -commutative. It follows that $\mathscr{X}{({R/{\mathfrak{m}}^{n+1}})}\to \mathscr{X}{({R/{\mathfrak{m}}^{n}})}$ is surjective for all $n \in{\mathbb{N}}$ , and it follows that $\lim _i \mathscr{X}{({R/{\mathfrak{m}}^{n+i}})}\to \mathscr{X}{({R/{\mathfrak{m}}^{n}})}$ is surjective. By effectivity of formal objects [SP23, Lemma 98.9.5, Tag 07X3] we have $\lim _i \mathscr{X}(R/{\mathfrak{m}}^{n+i}) = \mathscr{X}(\widehat{R})$ , where $\widehat{R}$ denotes the completion of $R$ . Thus if $R$ is complete we are done. Otherwise, Artin approximation [Reference ArtinArt69, Theorem 1.12], applied to the functor given by the isomorphism classes of objects of $\mathscr{X}$ , shows that the composition $\mathscr{X}(R) \to \mathscr{X}(\widehat{R}) \to \mathscr{X}(R/{\mathfrak{m}}^{n})$ is surjective for any $n$ , as required.

4.2 Weak approximation

Note that, unlike the case of varieties, the map $\mathscr{X}{({k})} \to \prod _v\mathscr{X}{({k_v})}$ need not be injective (its injectivity is related to triviality of various Tate–Shafarevich sets). Nevertheless we will sometimes abuse notation and use $\mathscr{X}{({k})}$ also to denote the image of this map, for instance in Lemma 4.10.

A rational map $f\colon \mathscr{X}_1 \dashrightarrow \mathscr{X}_2$ of finitely presented algebraic stacks is said to be birational if there exist dense open substacks $\mathscr{U}_i \subset \mathscr{X}_i$ such that $f$ is defined on $\mathscr{U}_1$ , $f(\mathscr{U}_1) \subseteq \mathscr{U}_2$ , and $f|_{\mathscr{U}_1}\colon \mathscr{U}_1 \to \mathscr{U}_2$ is an isomorphism. If such a rational map exists, we say that $\mathscr{X}_1$ and $\mathscr{X}_2$ are birationally equivalent.

Weak approximation for smooth stacks is a birationally invariant property.

In particular, if $\mathscr{X}$ admits an open dense substack $U$ that is isomorphic to a scheme, then weak approximation for $\mathscr{X}$ is equivalent to weak approximation for the scheme $U$ , and the definition does not offer anything new. Therefore, to get interesting new problems in general one should consider stacks with non-trivial generic stabilisers; such stacks typically admit open substacks that are gerbes over a scheme. In such cases one may hope to prove weak approximation using the following fibration result, which is a stacky version of [Reference Colliot-Thélène and GilleC-TG04, Proposition 1.1].

For neutral affine gerbes, weak approximation is equivalent to a statement in Galois cohomology.

Lemma 4.7. Let $G$ be a finite type affine group scheme over $k$ . Then $BG$ satisfies weak approximation if and only if the natural map

\begin{equation*} {\mathrm {H}}^1(k,G) \longrightarrow \prod _{v \in S} {\mathrm {H}}^1(k_v,G), \end{equation*}

is surjective for all finite sets of places $S$ of $k$ .

We emphasise that for a general algebraic stack $\mathscr{X}$ over $\mathbb{Q}$ it can happen that its coarse moduli space $\mathcal{X}$ satisfies weak approximation but $\mathscr{X}$ itself does not.

For $G=\mu _2$ , however, which is the case relevant to us, there is no such problem.

From Hensel’s lemma we obtain the following application of weak approximation.

Lemma 4.10. Let $\mathscr{X}$ be a smooth irreducible finitely presented algebraic stack over $k$ with $\prod _v\mathscr{X}(k_v) \neq{\varnothing }$ that satisfies weak approximation, and $S$ a finite collection of non-zero prime ideals of $k$ . Let $\mathscr{X}_{{\mathcal{O}}_k}$ be a model of $\mathscr{X}$ over ${\mathcal{O}}_k$ that is smooth over all elements of $S$ . Then the map

\begin{equation*} \mathscr {X}(k) \cap \prod _{\mathfrak {p} \in S} \mathscr {X}_{{\mathcal {O}}_k}({\mathcal {O}}_{\mathfrak {p}})\longrightarrow \prod _{\mathfrak {p} \in S} \mathscr {X}_{{\mathcal {O}}_k}({\mathbb {F}}_{\mathfrak {p}}), \end{equation*}

is surjective.

4.3 Weak approximation for $\mathscr{A}_2$

Before turning to $\mathscr{A}_3$ , we briefly consider the simpler case of $\mathscr{A}_2$ , where weak approximation holds. To see this, we first note that the natural map $\mathscr{M}_2 \to \mathscr{A}_2$ is birational. Indeed, the induced map on coarse moduli spaces is birational, and furthermore the map on generic stabilisers is an isomorphism: this is the strong Torelli theorem given (by Serre) in [Reference LauterLau01, Théorème 3].

Therefore, by Lemma 4.5 it suffices to prove that $\mathscr{M}_2$ satisfies weak approximation. However, an element of $\mathscr{M}_2(k_v)$ is represented by a hyperelliptic curve over $k_v$ , and one can approximate this to an arbitrary precision by a hyperelliptic curve over $k$ by simply approximating the coefficients (similarly for any finite collection of places).

A similar proof also shows that $\mathscr{A}_1$ satisfies weak approximation.

4.4 Weak approximation for $\mathscr{A}_3$

By Theorem1.5 we know that ${\mathcal{A}}_3$ is stably rational over $\mathbb{Q}$ , and hence satisfies weak approximation [Reference Colliot-Thélène and GilleC-TG04, Proposition 1.2].

Let us briefly consider how one would try to use this to prove that $\mathscr{A}_3$ satisfies weak approximation, and some of the subtleties that arise. Let $h\colon \mathscr{A}_3 \to{\mathcal{A}}_3$ be the coarse moduli map. Let $k$ be a number field and $S$ a finite set of places of $k$ . For $v \in S$ let $A_v \in \mathscr{A}_3(k_v)$ . Then there exists a $k$ -rational point $a \in{\mathcal{A}}_3(k)$ arbitrarily close to $a_v:=h(A_v)$ . The first issue that arises is that there is no guarantee that the fibre $h^{-1}(a)$ contains a rational point; in classical parlance $k$ is a field of moduli for $a$ , but there is no guarantee that $k$ is a field of definition for $a$ . Assuming that we overcome this issue and find $A \in \mathscr{A}_3(k)$ with $h(A) = a$ , the second issue is as follows: we have that $h(A)$ is close to each $h(A_v)$ , but this does not guarantee that $A$ is close to the $A_v$ ; in classical parlance this means that we can only guarantee that $A$ is close to some Galois twist of the $A_v$ , and not our original $A_v$ .

To overcome these issues we need to understand both the generic field of definition and the generic Galois twist. This is achieved by the following result of Shimura [Reference ShimuraShi72] reformulated in stacky language. (See [Reference Bresciani and VistoliBV24] for further applications of stacks to field of moduli questions.)

Proof of Theorem1.6

By Lemma 4.11 we know that $\mathscr{A}_3$ is birational to a neutral $\mu _2$ -gerbe over ${\mathcal{A}}_3$ . The latter satisfies weak approximation, and the fibres have rational points and satisfy weak approximation by Lemma 4.9. Thus the result follows from Lemmas 4.5 and 4.6.

Proof of Theorem1.1

This is immediate from Theorem1.6 and Lemma 4.10.

Proof of Theorem1.2

As $g \geq 7$ the coarse moduli space has general type. Let $p$ be a prime. To prove the result we will compare the cardinalities of the two sets

\begin{equation*} I_1={\mathrm {Im}}(\mathscr {A}_g({\mathbb {F}}_p) \longrightarrow {\mathcal {A}}_g({\mathbb {F}}_p)), \quad I_2={\mathrm {Im}}(\mathscr {A}_g({\mathbb {Q}}) \cap \mathscr {A}_g({\mathbb {Z}}_p) \longrightarrow {\mathcal {A}}_g({\mathbb {F}}_p)). \end{equation*}

The set $I_1$ is simply the collection of elements of ${\mathcal{A}}_g({\mathbb{F}}_p)$ that arise from some ppav over ${\mathbb{F}}_p$ . The set $I_2$ is the collection of elements of ${\mathcal{A}}_g({\mathbb{F}}_p)$ that arise as the reduction modulo $p$ of a ppav over $\mathbb{Q}$ with good reduction at $p$ (note that $I_2 \subseteq I_1$ ). We will show that $I_1$ has more elements than $I_2$ for all sufficiently large $p$ .

Firstly, the Lang–Weil estimates [Reference Lang and WeilLW54] imply that $|{\mathcal{A}}_g({\mathbb{F}}_p)| \sim p^m$ as $p \to \infty$ , where $m = g(g+1)/2 = \dim{\mathcal{A}}_g$ . By Lemma 4.11, spreading out, and the Lang–Weil estimates, we have

\begin{equation*} \# \{ a \in {\mathcal {A}}_g({\mathbb {F}}_p) \mid h^{-1}(a) \text { is not a $\mu _2$-gerbe}\} = O(p^{m-1}). \end{equation*}

However, a $\mu _2$ -gerbe over ${\mathbb{F}}_p$ is necessarily neutral: indeed, $\mu _2$ -gerbes are classified by ${{\rm H}}_{fppf}^2({\mathbb{F}}_p,\mu _2)$ [Reference OlssonOls16, Theorem 12.2.8]. But Kummer theory implies that ${{\rm H}}_{fppf}^2({\mathbb{F}}_p,\mu _2) = ({\rm Br}\,{\mathbb{F}}_p)[2]$ , which is trivial as ${\rm Br}\,{\mathbb{F}}_p = 0$ . We therefore deduce that $|I_1| \sim p^m$ as $p \to \infty$

For $I_2$ , as ${\mathcal{A}}_g$ has general type, the Bombieri–Lang conjecture [Reference PoonenPoo17, Conjecture 9.5.11] predicts that ${\mathcal{A}}_g({\mathbb{Q}})$ is not Zariski dense. Thus the Lang–Weil estimates imply that $|I_2| = O(p^{m-1})$ . Altogether we deduce that for all sufficiently large primes $p$ we have $|I_1| \gt |I_2|$ , which concludes the proof.