Arithmetic hyperbolicity and a stacky Chevalley–Weil theorem

We prove an analogue for algebraic stacks of Hermite–Minkowski's finiteness theorem from algebraic number theory, and establish a Chevalley–Weil type theorem for integral points on stacks. As an application of our results, we prove analogues of the Shafarevich conjecture for some surfaces of general type.


Introduction
Algebraic stacks are an important tool in modern algebraic geometry which naturally arise in the study of moduli problems. In this paper we study various arithmetic properties of stacks defined over number fields and their rings of integers. We obtain analogues for stacks of some classical theorems in algebraic number theory and arithmetic geometry, and give applications to the study of integral points on various moduli stacks.
1.1. Hermite-Minkowski for stacks. A classical result in algebraic number theory is the theorem of Hermite-Minkowski. A geometric way to phrase this is: given a positive integer n, a number field K, and a dense open subset U ⊂ Spec O K , the scheme U admits only finitely many isomorphism classes of finiteétale covers of degree at most n (versions of this are also known for Z-finitely generated subrings of C [20]).
Our first theorem is a generalisation of this result to algebraic stacks. One has to be careful however with formulating the correct statement. Firstly, as finite morphisms are by definition representable, to get interesting stacks one needs to study propeŕ etale morphisms. Secondly, extra phenomena appear in the case of stacks due to their inertia groups (see §2). For example, if X → Y is a properétale morphism, then its degree deg(X → Y ) is only a rational number in general. Thirdly, bounding the degree is not sufficient: for every positive integer n, the disjoint union of n-copies of the classifying stack BZ/nZ is a properétale cover of Spec Z of degree 1. One therefore also needs to bound what we call the inertia degree Ideg(X/Y ) (see §2 for a precise definition). Bearing these considerations in mind, our result is the following. Theorem 1.1 (Hermite-Minkowski for stacks). Let A ⊂ C be a Z-finitely generated subring with fraction field K := Frac(A), and let n ∈ N. Then the set of K-isomorphism classes of properétale algebraic stacks X over A such that max{deg(X/A), Ideg(X/A)} ≤ n is finite. Theorem 1.1 is proved by reducing to two cases: finiteétale morphisms, where this is the usual Hermite-Minkowski, and the case of gerbes, which we handle using non-abelian cohomology.
Note that Theorem 1.1 may be viewed as a common generalisation of the theorem of Hermite-Minkowski, and the fact that the n-torsion of Br O k [S −1 ] is finite for any n and any finite set of finite places S (this follows by viewing Brauer group elements as gerbes). The latter finiteness is well-known from class field theory and can be deduced from the fundamental exact sequence [40,Ex. 6.9.4] .
1.2. Chevalley-Weil for stacks. The Hermite-Minkowski theorem has many applications in number theory, and we expect our version for stacks to have similar applications. We give one such here, which is a version of the Chevalley-Weil theorem for stacks; see [10] or [46, §4.2] for a formulation of the classic Chevalley-Weil theorem.
Theorem 1.2 (Chevalley-Weil for stacks). Let A ⊂ C be an integrally closed Zfinitely generated subring, and let Y be a finite type separated Deligne-Mumford algebraic stack over A. Let X → Y be a morphism such that X C → Y C is properétale. Suppose that for every Z-finitely generated subring A ⊂ A ′ ⊂ C the groupoid X(A ′ ) is finite. Then the groupoid Y (A) is finite.
In the classical version of Chevalley-Weil, one takes A = O k [S −1 ], the ring of Sintegers of some number field k, and X and Y are models of some varieties over O k . The conclusion is that Y has only finitely many integral points, provided X has only finitely many O L [T −1 ]-integral points for every finite field extension L/k and every finite set of finite places T of L.
Our stacky version (Theorem 1.2) also concerns integral points. However, for stacks, Y (A) is a groupoid in general, and not necessarily a set; thus one needs to quotient out by isomorphisms to get meaningful finiteness statements (we say that a groupoid G is finite if the set of isomorphism classes of objects of G is finite).
Many stacks naturally arise as moduli stacks of varieties. If Y has such a modular interpretation, then the finiteness of Y (A) can be interpreted in terms of analogues of the Shafarevich conjecture on the finiteness of varieties with good reduction outside a given set of places. (This was originally proved by Faltings for curves and abelian varieties in [15,16]. ) To explain the connection between finiteness of integral points on stacks to the aforementioned Shafarevich conjecture more precisely, let g ∈ N and let A g be the stack of principally polarised abelian schemes over Z. For a number field k and a finite set of places S of k, an object A in A g (O k [S −1 ]) is a principally polarised abelian scheme over O k [S −1 ]; the corresponding abelian variety A k over k therefore has good reduction outside of S. It is easy to see that the essential image of the natural morphism of groupoids consists of exactly those principally polarised abelian varieties of dimension g over k with good reduction outside of S. In particular, an equivalent way to state Faltings's finiteness theorem is that the image of (1.1) is finite. (In fact the groupoid A g (O k [S −1 ]) itself is also finite, as the functor (1.1) is fully faithful due to the properties of the Néron model of an abelian variety.) Thus, as this example shows, it is very natural to study integral points on stacks in arithmetic geometry.
1.3. Applications. As an application of our results, we obtain a finiteness result for certain surfaces of general type. By arguing directly with the stack, the proof of the following result becomes an application of classification results due to Beauville [13,Appendix] for such surfaces and our stacky Chevalley-Weil theorem. Theorem 1.3. Let A ⊂ C be an integrally closed Z-finitely generated subring and q ≥ 4. Then the set of A-isomorphism classes of smooth proper surfaces X over A with ω X/A relatively ample and p g (X C ) = 2q(X C ) − 4 is finite.
Other applications of the Chevalley-Weil theorem (Theorem 1.2) arise in the case of moduli stacks which are properétale gerbes over other moduli stacks. Such situations arise quite naturally, and we give various applications of the stacky Chevalley-Weil theorem to such moduli stacks in [28].
The stacky Chevalley-Weil theorem is part of a more general study in this paper of the phenomenon of arithmetic hyperbolicity for stacks. A variety over an algebraically closed field k of characteristic zero is called arithmetically hyperbolic over k if, for every Z-finitely generated subring A ⊂ k and every finite type separated scheme X over A with X k ∼ = X, the set of A-valued points X (A) on X is finite. In §4, we extend this natural notion to algebraic stacks and prove several basic properties. Further properties of arithmetically hyperbolic varieties are obtained in [21], building on the results in §4.
Outline of the paper. In §2 we define and study the basic properties of the degree and inertia degree of a morphism of stacks, together with some properties of propeŕ etale morphisms. In §3 we prove Theorem 1.1 and §4 develops the general theory of arithmetic hyperbolicity for algebraic stacks. Theorem 1.2 is proved in §5, and §6 contains applications of our results, including the proof of Theorem 1.3 and a purely "transcendental" criterion for a stack to be arithmetically hyperbolic (Theorem 6.4).
Further applications of the Chevalley-Weil theorem and transcendental techniques will appear in a forthcoming paper [28]. Vistoli for helpful discussions. The first-named author gratefully acknowledges support of SFB/Transregio 45. The second-named author is supported by EPSRC grant EP/R021422/1.
Conventions. For a number field K, we let O K denote its ring of integers. If S is a finite set of finite places of K, If k is a field, a variety over k is a finite type separated k-scheme.
An arithmetic scheme is a finite type flat scheme over Z.
We abbreviate quasi-compact quasi-separated as qcqs. A (small) groupoid G is finite if the set π 0 (G) of isomorphism classes of objects in G is a finite set. (Note that we do not ask for the automorphism group of an object of G to be finite.) If f : X → Y is a morphism of algebraic stacks, then we say that f is quasi-finite if f is finitely presented and locally quasi-finite (in the sense of [48,Tag 06PU]). If f : X → Y is quasi-finite, for all geometric points y : Spec k → Y of Y , the groupoid X y (k) is finite.
Concerning gerbes, we follow the (standard) conventions of the stacks project [48, Tag 06QB]. If G is a locally finitely presented flat group scheme over a scheme S, we let BG = [S/G] be the classifying stack of G-torsors over S (for the fppf topology). For a scheme T over S, the objects of the groupoid BG(T ) are G T -torsors over T . In particular, there is a natural bijection π 0 (BG(T )) = H 1 (T, G T ). (All cohomology in this paper is taken with respect to the fppf topology).
For X a stack, we let I X be the (absolute) inertia stack of X. For X → Y a morphism of stacks, we let I X/Y be the relative inertia stack of X → Y . We refer the reader to [48, Tag 050P] and [48, Tag 04YX] for precise definitions.
For an (abstract) group G and a scheme S, we denote the associated constant group scheme over S by G S . Let X be a finitely presented algebraic stack over a field k. Let A ⊂ k be a subring. A model for X over A is a pair (X , φ) with X a finitely presented algebraic stack over A and φ : X × A k → X an isomorphism over k. We will usually omit φ from the notation.
A morphism X → Y of algebraic stacks is called representable (resp. strongly representable) if for every scheme Z and morphism Z → Y , the fibre product Z × Y X is an algebraic space (resp. a scheme). We use slightly different terminology to the stacks project [ where the sum is over the finitely many irreducible components Y ′ of Y red . With these conventions, roughly speaking, the degree of a morphism to an integral scheme is defined to be the degree of its generic fibre.
Let now f : X → Y be a finitely presented Deligne-Mumford morphism of qcqs algebraic stacks. Let V → Y be a smooth surjective finitely presented morphism from a scheme. By definition [48, Tag 04YW], the fibre product X × Y V is a finitely presented Deligne-Mumford algebraic stack over (the scheme) V . If f is representable, then X × Y V is an algebraic space, hence from the above we may define If general (when f is not necessarily representable), choose a scheme U and a finitely presentedétale surjective morphism U → X × Y V . We may then define as each morphism occurring on the right is representable. In the special case of a separated dominant morphism of finite type qcqs integral Deligne-Mumford algebraic stacks, our definition recovers Vistoli's definition [52, Def. 1.15] (note however that for Vistoli, "representable" means "strongly representable"). A similar argument to [52,Lem. 1.16] shows that this definition is independent of the above choices, and that given finitely presented Deligne-Mumford morphisms X → Y and Y → Z of qcqs stacks with Z integral, we have the multiplicative property Our definition of inertia degree is the following. Here I X/Y denotes the relative inertia stack. Note that I X/Y → X is a finitely presented unramified quasi-finite representable morphism, hence Deligne-Mumford. The inertia degree "measures" how far a morphism is from being representable over some dense open substack of Y .
2.3. Properétale morphisms. We now study the above notions of degree and inertia degree for a properétale morphism X → Y of algebraic stacks.
Lemma 2.2. Let f : X → Y be a properétale morphism of algebraic stacks. Then the relative inertia stack I X/Y is finiteétale over X.
Proof. Since X and X × Y X are properétale over Y , the diagonal ∆ : [48,Tag 0CIR]. As the pull-back X × ∆,X× Y X,∆ X of ∆ along itself is the relative inertia stack I X/Y of X → Y (see [48, Tag 034H]), the relative inertia stack I X/Y of X → Y is properétale over X. As I X/Y → X is representable [48, Tag 050P] and properétale, it is therefore strongly representable by Knutson's criterion [29,Cor. 6.17], hence finiteétale as claimed. Proof. Assume BG → S is properétale, so that the inertia stack I BG → BG is finiteétale (Lemma 2.2). Consider the canonical morphism S → BG = [S/G] corresponding to the trivial torsor G → S. Let Aut G (G) be the automorphism group of the trivial G-torsor G → S (as an object in the category of G-torsors over S). By definition, we have a Cartesian diagram Thus, the morphism Aut G (G) → S is finiteétale as it is the pull-back of a finité etale morphism. However, as the automorphism group Aut G (G) of the trivial (left) G-torsor is isomorphic to G over S, the morphism G → S is finiteétale. If G → S is finiteétale, S → [S/G] = BG is finiteétale. Therefore, as the composition S → BG → S is the identity, the morphism BG → S isétale. Moreover, as the morphism BG → S is a coarse space map, it is proper [43,Thm. 6.2]. This shows that BG → S is proper andétale.
To study the degree and inertia degree in families, we will use the following rigidification result for stacks (cf. [48, Tag 04V2]). Lemma 2.4. Let f : X → Y be a properétale morphism of algebraic stacks. Assume that Y is a scheme. Then the morphism f factorises as where X → R is a properétale gerbe and R → Y is a finiteétale morphism of schemes.
Proof. By Lemma 2.2, the relative inertia stack I X/Y → X is finiteétale. By standard rigidification results [2, Thm. A.1] (but see also for instance [1,Thm. 5.1.5] or [42,Thm. 5.1]), there exist an algebraic stack R over Y (denoted by R = X I X ), a propeŕ etale gerbe X → R (given by the "rigidification" X → X I X ), and a representable morphism R → Y such that the morphism in question factorises as X → R → Y . To conclude the proof, it suffices to show that R → Y is a finiteétale morphism of schemes (not that a priori R is just an algebraic space).
Firstly, note that the morphism X → R is surjective, as it is a gerbe. Now, since X → R is a finitely presentedétale surjective morphism and beingétale is local on the source, it follows that R → Y isétale.
We now show that R → Y is separated. To prove this, we use the valuative criterion. (Note that R → Y is finitely presented, hence quasi-separated). Let O K be a valuation ring with fraction field K. Let f 1 : Spec O K → R and f 2 : Spec O K → R be morphisms which give rise to the same O K -point of Y and which agree generically, i.e., f 1,K = f 2,K . We need to show that f 1 = f 2 . As X → R is a properétale gerbe, there exists a finite field extension L/K with O L the integral closure of O K in L and morphisms g i : Spec O L → X such that g 1,L = g 2,L and such that the diagram [33,Ex. 4.10.2] and finite). Therefore, as algebraic spaces are sheaves for the fppf topology, we find that As R → Y is separated andétale, Knutson's criterion [29,Cor. 6.17] implies that R is a scheme. Now, as R → Y is a finitely presented separated morphism of schemes and X → Y is a proper morphism with X → R surjective, it follows that R → Y is proper [48,Tag 0CQK]. We see that R → Y is a properétale morphism of schemes, hence finiteétale. This concludes the proof.
We now show that the degree and inertia degree of a properétale morphism are constant along the fibres over an integral base, as is familiar in the case of schemes. Lemma 2.5. Let X → Y be a properétale morphism of qcqs algebraic stacks with Y integral. Let k be a field and let y ∈ Y (k). Then Proof. As the statement of the lemma is local on Y for the fppf topology, we may assume that Y is an integral scheme. Moreover, by summing over the irreducible components of X, we may assume that X is integral. We apply rigidification to X → Y (Lemma 2.4) to see that X → Y factors as X → R → Y with X → R a properétale gerbe and R → Y a finiteétale morphism of integral schemes. As R → Y is representable (thus has inertia degree 1), the stacks I X/Y and I X/R are isomorphic over X, so that we have deg(I X/R /X) = deg(I X/Y /X). Therefore, by (2.1) and Lemma 2.3 we have Similarly, as the properétale morphism X y → Spec k rigidifies as X y → R y → Spec k, we have As R → Y and I X/Y → X are both finiteétale (hence strongly representable) with integral codomain, it thus suffices to prove the result when X → Y is a finiteétale morphism of schemes with Y integral; however this case is well-known and follows from the fact that the push-forward of O X to Y is locally free of rank deg(X/Y ). This proves the constancy of the degree. For the inertia degree, by (2.1) we have whence the result here follows from the previous case.

Hermite-Minkowski for algebraic stacks
In this section we prove our version of Hermite-Minkowski's theorem for algebraic stacks (Theorem 1.1). We begin with some results on gerbes.
A band over a scheme S is an S-object of the stack of bands, as defined in [19, Déf. IV.1.1.6]. (Band is the current accepted English translation of the french lien). If G is a group scheme over S, then we let lien(G) be the associated band over S. One can associate to every gerbe over S a band over S [19,§IV.2.2].
In general the band of a gerbe over S is not necessarily of the form lien(G) for some G (see Remark 3.2). However, for a properétale gerbe, we have the following weaker statement.
Lemma 3.1. Let S be an integral scheme and X → S a properétale gerbe. There is a finite (abstract) group G with #G = Ideg(X/S) such that the band of X → S is isomorphic to lien(G S ) in the category of bands over S, locally for theétale topology on S.
Proof. Let k be an algebraically closed field, and let x be an object of X(k). Let G be a finite (abstract) group such that the inertia group I x of x is isomorphic to the constant group scheme G k over k.
Since X → S is proper andétale, it follows that X is Deligne-Mumford over S. Therefore, there is a scheme U and a finitely presentedétale surjective morphism U → X. Consider the Cartesian diagrams In particular, the morphism X × S U → U is a properétale neutral gerbe. Therefore, there is a flat finitely presented group scheme G over U such that By definition, as U → S is anétale (hence fppf) covering of S and X × S U ∼ = BG over U, it follows that the band of X → S is isomorphic to lien(G) over U. Moreover, since X → S is properétale, the morphism BG → U is properétale (with the same inertia degree as X → S). By Lemma 2.3, the group scheme G → U is finité etale of degree Ideg(X/S). To complete the proof, we now "trivialize" the group scheme G over U. More precisely, let V → U be a finitely presentedétale (but not necessarily surjective) morphism with V a non-empty connected scheme such that G V is constant over V . Then, as X → S is a gerbe, it follows that X V ∼ = BG V . However, as the inertia group of any object of X(V ) is an inner form of G V , we also have that X V ∼ = BG V . Since X V → V is properétale with inertia degree Ideg(X/S), it follows that BG V → V is properétale with inertia degree Ideg(X/S). It follows from Lemma 2.3 that G V has degree Ideg(X/S) over V , i.e., #G = Ideg(X/S).
Let L be the band of Thus, as L V ∼ = lien(G) V , we conclude that L is isomorphic to lien(G) over S, locally for theétale topology on S.
Remark 3.2. If X → S is a properétale abelian gerbe, then there is an abelian group G such that the band of X is isomorphic to lien(G) over S. However, this is not necessarily true if X → S is not abelian, as the following example shows. (This example was communicated to us by Andrew Kresch).
Let Z/4Z act as the automorphism group of Z/5Z and let G = Z/5Z ⋊ Z/4Z be the corresponding semidirect product. There is an exact sequence of groups Let K be an imaginary quadratic extension of Q and consider X := [Spec(K)/G], where G acts through Z/2Z = Gal(K/Q). Note that X is the gerbe of "lifts of the Z/2Z-torsor K/Q to a G-torsor". This is a gerbe over S := Spec Q, and it isétale locally isomorphic to B(D 5 ). The outer automorphism group of D 5 is Z/2Z, so the band splits over a quadratic extension of Q, namely K. Suppose that there is a finité etale group scheme H over Q such that the band of X is isomorphic to lien(H). Then H is a form of D 5 (corresponding to an element in H 1 (Q, Aut(D 5 )) = H 1 (Q, G)), and there would have to be a continuous homomorphism Gal(Q) → Aut(D 5 ) = G such that the composite Gal(Q) → G → Z/2Z corresponds to the quadratic extension K. In particular, this would induce a homomorphism Gal(Q) → Z/4Z → Z/2Z, contradicting the fact that, for any cyclic degree 4 extension of Q, the intermediate field is always real quadratic (the discriminant of the quadratic extension being a sum of two squares). Thus the band of the properétale gerbe X is not isomorphic to the band induced by a group scheme over Q.
We next recall how to classify bands over S which are locally isomorphic to lien(G). Lemma 3.4. Let S be an integral scheme. Let X → S be a properétale gerbe. Let L be its band over S. Then the center Z := Z(L) of the band L is a finiteétale group scheme over S of degree dividing Ideg(X/S), and the set of L-equivalence classes of properétale gerbes over S banded by L is in bijection with H 2 (S, Z).
Proof. By Lemma 3.1, there is a finite group G of order Ideg(X/S) such that L is isomorphic to lien(G), locally for the fppf topology on S. In particular Z(L) is locally isomorphic to lien(Z(G)), so that Z(L) is a finiteétale group scheme over S of degree #Z(G). Since #Z(G) divides #G = Ideg(X/S) this shows the first statement.
To prove the second statement, note that every gerbe over S banded by L is propeŕ etale over S. (Indeed, by descent it suffices to show this holds for a neutral gerbe Y → S banded by L. However, the band of Y → S is isomorphic to lien(G) over S, so that Y ∼ = BG. Therefore, Y → S is properétale by Lemma 2. 3.2. Hermite-Minkowski for gerbes. We now prove a version of Theorem 1.1 for gerbes. We require the following finiteness statements for cohomology sets.
Lemma 3.5. Let B be an integral arithmetic scheme and G a finiteétale group scheme over B. The following statements hold.
(1) The pointed set H 1 (B, G) is finite. Assume further that G is abelian.
(2) If dim B = 1, then the set H r (B, G) is finite for all r ≥ 0.
(3) If the degree of G over B is invertible on B, then the set H r (B, G) is finite for all r ≥ 0.
Proof. As G is finiteétale, by Hermite-Minkowski for arithmetic schemes [20] there are only finitely many possibilities for the scheme underlying the torsor representing each element of H 1 (B, G). As there are also only finitely many possibilities for the Gaction on such a scheme, this proves the finiteness of H 1 (B, G Our main finiteness result is for gerbes of fixed inertia degree over arithmetic schemes, and reads as follows. Proof. We prove the result using the classification results from the previous section. Let X → B be a properétale gerbe (for the fppf topology) of inertia degree e, and note that X is integral. By Lemma 3.1, there is a finite (abstract) group G of order e such that the band of X → B is locally isomorphic to lien(G) in the category of bands. By Lemma 3.3, the band of X → B is canonically an object of H 1 (B, Out(G)). Note that, as Out(G) is finiteétale group scheme over B, the latter set is finite (Lemma 3.5). Thus, as there are only finitely many finite groups of order e, there are finitely many bands L 1 , . . . , L r over B such that any properétale gerbe of inertia degree e is banded by some L i .
Fix i ∈ {1, . . . , r} and define L := L i . Note that the center Z(L) of L is a finité etale group scheme over B and that the set of L-equivalence classes of properétale gerbes over B banded L is in bijection with H 2 (B, Z(L)); see Lemma 3.4. Since Z(L) is abelian, the group H 2 (B, Z(L)) is finite under our assumptions (Lemma 3.5).
From the above we conclude that the set of pairs (L, X) with L a band over B and X a (properétale) gerbe banded by L of inertia degree e over B is finite, up to equivalence as a pair. However, the latter (finite) set surjects onto the set of Bisomorphism classes of properétale gerbes over B of inertia degree e, so that the latter set is finite, as required. , it follows that X and X − are isomorphic as algebraic stacks (hence gerbes) over S.

3.3.
Hermite-Minkowski for properétale morphisms. If S is a scheme and q is a non-zero rational number, we say that q is invertible on S if, for every prime number p with ord p (q) = 0, we have that p is invertible on S. The following result generalizes Theorem 3.7 from properétale gerbes to properétale morphisms. Theorem 3.9. Let B be an integral arithmetic scheme. Let n be a positive integer and let q be a positive rational number. If dim B > 1, assume that q is invertible on B. Then the set of B-isomorphism classes of properétale morphisms X → B of degree q such that Ideg(X/B) ≤ n is finite.
Proof. By the rigidification lemma (Lemma 2.4), for any properétale morphism X → B as in the statement, there is a scheme R such that the morphism X → B factors as X → R → B, where X → R is a properétale gerbe and R → B is finiteétale. As R → B is finiteétale, it follows that R is an arithmetic scheme. We are first going to show that the set of B-isomorphism classes of B-schemes R appearing above is finite. Now, by ( [16, p. 209] or [20]). Similarly, as Ideg(X/R) ≤ n is bounded, the set of the R-isomorphism classes of properétale gerbes X → R occuring above is finite by our version of Hermite-Minkowski for properétale gerbes (one applies Theorem 3.7 to the irreducible components of R). This implies that the set of B-isomorphism classes of properétale gerbes X → R occuring above is also finite. This completes the proof.
We now prove the stacky version of Hermite-Minkowski formulated in the introduction.
Proof of Theorem 1.1. Using (3.1) we see that the denominator of deg(X/A), when expressed as a/b with a and b coprime, divides Ideg(X/A). As both Ideg(X/A) and deg(X/A) are bounded, it follows that the height H(deg(X/A)) is bounded, thus deg(X/A) ranges over a finite set of rational numbers. As the statement is up to K-isomorphism (as opposed to A-isomorphism), replacing Spec A by an affine open if necessary, we may assume that deg(X/A) is invertible in A. The result now follows from Theorem 3.9.

Arithmetic hyperbolicity
In this section we extend the notion of arithmetic hyperbolicity (employed for instance in [50, §2], and [3,4]) to algebraic stacks, and establish its formal properties. Throughout this section k is an algebraically closed field of characteristic zero. Our aim is to extend this natural property of a variety over Q to algebraic stacks defined over our algebraically closed field k (which might not be Q).
Let A ⊂ k be a Z-finitely generated subring and let X be a finitely presented algebraic stack over A. As the A-valued points X (A) on X form a groupoid (which is not naturally a set in general), to study the finiteness of integral points, we consider the set π 0 (X (A)) of isomorphism classes of objects in X (A). Moreover, the most flexible notion of arithmetic hyperbolicity is obtained by considering the image of π 0 (X (A)) in the set of "geometric" points π 0 (X (k)). This "flexibility" will help in establishing some of the basic geometric properties we require. For numerous applications it is also useful to consider the image of the integral points inside the rational points (cf. the map (1.1)).

Definition 4.1. [Arithmetic hyperbolicity]
A finitely presented algebraic stack X over an algebraically closed field k is arithmetically hyperbolic (over k) if there exist a Z-finitely generated subring A ⊂ k and a model X of X over A such that, for all Z-finitely generated subrings A ′ ⊂ k containing A, the set Im[π 0 (X (A ′ )) → π 0 (X (k))] is finite.

Remark 4.2.
If X is a variety over k, then X is arithmetically hyperbolic over k if and only if there is a Z-finitely generated subring A ⊂ k and a model X of X over A such that X is a separated scheme and, for all Z-finitely generated subrings A ′ ⊂ k containing A, the set X (A ′ ) is finite. (This follows from Lemma 4.8 and the fact that π 0 (X (A)) = X (A) injects into π 0 (X(k)) = X(k).) Consider also the important case k =Q. Then a Z-finitely generated subring A ⊂Q is an order in some number field. However in the study of arithmetic hyperbolicity, we are free to replace Spec A by a dense open subset, as it only makes the problem more difficult, so we may assume that Spec A is actually regular. We deduce that X is arithmetically hyperbolic if and only if there is a number field K and some model X for X over O K , such that X (O L [S −1 ]) is finite for all number fields K ⊂ L and all finite sets of places S of L. A more general version of this statement is provided by Lemma 4.9.
Remark 4.3. The fact that we work with stacks leads to some pathologies that are worth keeping in mind. For instance, a rational point on an algebraic stack can come from infinitely many pairwise non-isomorphic integral points. Indeed, there is a Z-finitely generated subring A ⊂ C such that the map Pic(A) = π 0 (BG m (A)) → π 0 (BG m (FracA)) = Pic(Frac(A)) = {1} has infinite fibres. The problem here is that the stack BG m is non-separated.
Another phenomenon is that infinitely many rational points can give rise to the same geometric point. Namely, consider Bµ 2 over Q. We have π 0 (Bµ 2 (Q)) = Q * /Q * 2 , yet π 0 (Bµ 2 (Q)) is a singleton. It is for these reasons that we consider the image of the integral points inside the geometric points in Definition 4.1.
However, for finitely presented separated Deligne-Mumford stacks, being arithmetically hyperbolic is equivalent to the more natural and a priori stronger condition of having only finitely many (isomorphism classes of) integral points; see Theorem 4.18 for a precise statement.
Remark 4.4. Note that Z acts on A 1 k via translation. Since the action is free, the algebraic stack X = [A 1 /Z] is an algebraic space. Note that X is of finite type over C. However, by [48, Tag 06Q2], the stack X is not finitely presented over C, as its diagonal is not quasi-compact. However, note that a finite type algebraic stack over C with affine diagonal (or quasi-compact diagonal) is in fact finitely presented (by definition) over C. Thus, the distinction between finite type and finitely presented only appears when the stack in question is "highly" non-separated. Example 4.6. Let g ≥ 2 be an integer. Let M g be the finite type separated Deligne-Mumford algebraic stack of smooth proper curves of genus g over Z. Then M g,k is arithmetically hyperbolic over k.
Remark 4.7. Let A be an abelian variety over C, and let X ⊂ A be a closed subvariety. Then X is arithmetically hyperbolic over C if and only if X is does not contain a translate of an abelian subvariety of A. This is a consequence of Faltings's theorem [17]. Generalizations to subvarieties of semi-abelian varieties were obtained by Vojta [54,55], and applications of these results to certain ball quotients were given by Yeung [56]. Non-trivial affine examples of arithmetically hyperbolic varieties are given in [3,4,11,12,17,32,53].

Basic properties of arithmetically hyperbolic stacks.
Lemma 4.8 (Independence of model). Let X be a finitely presented arithmetically hyperbolic algebraic stack over k. Then, for all Z-finitely generated subrings B ⊂ k and all models Y for X over B, the set Im[π 0 (Y(B)) → π 0 (Y(k))] is finite.
Proof. Since X is arithmetically hyperbolic, there exist a Z-finitely generated subring A ⊂ k and a model X of X over A such that, for all Z-finitely generated subrings A ′ ⊂ k containing A, the set Im[π 0 (X (A ′ )) → π 0 (X (k))] is finite. Now, let B ⊂ k and Y be as in the statement of the lemma. Note that there is a Z-finitely generated subring C ⊂ k containing A and B such that X C ∼ = Y C . It follows that Im[π 0 (Y(A)) → π 0 (Y(k))] is a subset of Im[π 0 (Y(C)) → π 0 (Y(k))] = Im[π 0 (X (C)) → π 0 (C(k))].
As the latter set is finite, this concludes the proof. Lemma 4.9 (Can test on smooth subrings). Let X be a finitely presented algebraic stack over k. Then X is arithmetically hyperbolic over k if and only if for every smooth Z-finitely generated subring A ⊂ k and model X for X over A ⊂ k the set Im[π 0 (X (A)) → π 0 (X (k))] is finite.
Proof. Let A ⊂ k be a Z-finitely generated subring and let X be a model for X over A. Let A ⊂ A ′ ⊂ k be a Z-finitely generated subring. Let A ′ ⊂ A ′′ ⊂ k be a smooth Z-finitely generated subring. Then the set Im[π 0 (X (A ′′ )) → π 0 (X (k))] is finite by assumption. This implies that the subset is finite, and concludes the proof.
Remark 4.10. If f : X → Y is a (finitely presented) morphism of finitely presented algebraic stacks over k, then there is a Z-finitely generated subalgebra A ⊂ k, and (finitely presented) morphism F : X → Y of finitely presented algebraic stacks over A such that F k ∼ = f .
The following simple lemma can be viewed as confirming the intuitive statement that a "space which fibers in hyperbolic spaces over a hyperbolic variety" is hyperbolic.
Lemma 4.11 (Fibration property). Let Y → Z be a morphism of finitely presented algebraic stacks over k. If Z is arithmetically hyperbolic and, for all geometric points z : Spec k → Z of Z, the algebraic stack Y z is arithmetically hyperbolic, then Y is arithmetically hyperbolic.
Proof. We first use Remark 4.10 and the arithmetic hyperbolicity of Z over k to spread out Y → Z over some A ⊂ k. More precisely, let A ⊂ k be an integrally closed Z-finitely generated subring and let Y → Z be a morphism of finitely presented algebraic stacks over A which is isomorphic to Y → Z over k such that, for all Zfinitely generated subrings A ′ ⊂ k containing A, the set is finite. Let A ′ ⊂ k be a Z-finitely generated subring. To prove the lemma, it suffices to show that the set Im[π 0 (Y(A ′ )) → π 0 (Y(k))] is finite. To do so, consider the natural morphism of sets Let z : Spec k → Z be a point in the image of the latter morphism of sets, and let z : Spec A → Z be an extension of z over A. Since Im[π 0 (Z(A ′ )) → π 0 (Z(k))] is finite, it suffices to show that the fibre over z is finite. However, the fibre over z is contained in the set Since Y z is arithmetically hyperbolic and Y z is a model for Y z , the latter set is finite (Lemma 4.8).
Lemma 4.12. Let σ : k → L be a morphism of algebraically closed fields of characteristic zero. Let X be a finitely presented algebraic stack over k. If X L = X ⊗ k,σ L is arithmetically hyperbolic over L, then X is arithmetically hyperbolic over k.
Proof. If A ⊂ k is a Z-finitely generated subalgebra A ⊂ k, then σ(A) ⊂ L is a Zfinitely generated subalgebra of L. This observation easily allows one to conclude.
If σ is an element of Aut(k) and X is an algebraic stack over k, we let X σ be the algebraic stack X × k,σ k. Lemma 4.13 (Conjugation property). Let σ be a field automorphism of k. Let X be a finitely presented algebraic stack over k. If X is arithmetically hyperbolic over k, then X σ is arithmetically hyperbolic over k.
Proof. This follows from Lemma 4.12.
Lemma 4.14. Let X → Y be a morphism of finitely presented algebraic stacks over k. If Y is arithmetically hyperbolic and X → Y is a gerbe, then X is arithmetically hyperbolic.
Proof. If B is a gerbe over k, then π 0 (B(k)) is a singleton. Thus, a gerbe over k is arithmetically hyperbolic over k. Therefore, for all y in Y (k), as the fibre X y is a gerbe over k, we see that the fibres of X → Y are arithmetically hyperbolic. Therefore, the lemma follows from the fibration property (Lemma 4.11).
Lemma 4.15. Let X be a finitely presented algebraic stack over k, and let X red be the associated reduced algebraic stack. The algebraic stack X is arithmetically hyperbolic if and only if X red is arithmetically hyperbolic.
Proof. Note that the fibres of the morphism X red → X are arithmetically hyperbolic. In particular, if X is arithmetically hyperbolic, then X red is arithmetically hyperbolic (Lemma 4.11). Conversely, assume that X red is arithmetically hyperbolic. Let A ⊂ k be a finitely generated Z-algebra and let X be a model for X over A. Since A is an integral domain, every morphism Spec A → X factors uniquely via X red . Thus, the natural map of sets Im[π 0 (X red (A)) → π 0 (X red (k))] → Im[π 0 (X (A)) → π 0 (X (k))] is surjective. Since X red is arithmetically hyperbolic, the set Im[π 0 (X red (A)) → π 0 (X red (k))] is finite (Lemma 4.8). We conclude that X is arithmetically hyperbolic.
Proposition 4.16. Let Y → Z be a quasi-finite morphism of finitely presented algebraic stacks over k. If Z is arithmetically hyperbolic, then Y is arithmetically hyperbolic.
Proof. By Lemma 4.11, it suffices to show that the k-fibres of Y → Z are arithmetically hyperbolic. However, as the k-fibres are quasi-finite (finitely presented) algebraic stacks over k, they are clearly arithmetically hyperbolic.
Lemma 4.17. Let X be a finitely presented algebraic stack over k. Then X is arithmetically hyperbolic if and only if all irreducible components of X are arithmetically hyperbolic.
Proof. Suppose that X is arithmetically hyperbolic. Let Z be an irreducible component of X. Since Z → X is quasi-finite, it follows that Z is arithmetically hyperbolic (Proposition 4.16). The converse follows from the fact that X has only finitely many irreducible components.

4.3.
The twisting lemma. The definition of arithmetic hyperbolicity is inherently "geometric", as it is a condition on k-points, with k algebraically closed. However, we show in this section that under certain assumptions on the model (e.g., separated with affine diagonal), we can in fact deduce finiteness results for A-valued points. We refer to this result as the "twisting lemma (for arithmetic hyperbolicity)".
The twisting lemma below generalizes our previous results for canonically polarized varieties [22,Lem. 4.1], complete intersections [25,Thm. 4.10], and certain Fano varieties [27,Prop. 4.7]. It is very useful in applications; it says that, in certain cases, to deduce finiteness of (isomorphism classes of) integral points, it suffices to prove the a priori weaker claim that X is arithmetically hyperbolic. This latter property can be tackled using more geometric techniques. We give such applications in §6. (1) The stack X is arithmetically hyperbolic over k.
(2) For all Z-finitely generated subrings A ⊂ k with fraction field K and all models X → Spec A for X over A, the set Im[(π 0 (X (A)) → π 0 (X (K))] is finite. (3) For all Z-finitely generated integrally closed subrings A ⊂ k and all models X → Spec A for X over A with X a separated algebraic stack over A with affine diagonal (e.g. X is a separated Deligne-Mumford stack over A), the set π 0 (X (A)) of isomorphism classes of A-integral points on X is finite.
Proof. To show that (1) implies (2), let X be as in (2) and assume X = X k is arithmetically hyperbolic over k. Then Im[(π 0 (X (A)) → π 0 (X (k))] is finite. We have to show that the image inside the K-points is also finite. Since X is finitely presented separated Deligne-Mumford over k and Spec k → Spec K is fpqc, the finitely presented stack X K over K is separated and Deligne-Mumford stack over K. To prove the finiteness of Im[(π 0 (X (A)) → π 0 (X (K))], we may replace Spec A by a dense open. Therefore, we may and do assume that Spec A is smooth over Z. Since X → Spec A is finitely presented, by spreading out of separatedness and Deligne-Mumfordness [44,Prop. B.3], we may and do assume that X is finitely presented, separated, and Deligne-Mumford over A. (In particular, the inertia group scheme of an object x in X (A) is finite unramified over A.) Let x be an object of X (A), and write x K and x k for the corresponding objects in X (K) and X(k), respectively. To show that (1) implies (2), it suffices to show that the set of K-isomorphism classes of objects y in X (A) such that y k is isomorphic to x k is finite. As X K is finitely presented separated Deligne-Mumford over K, the inertia group scheme I x K of x K ∈ X (K) is finiteétale over K. As we may replace Spec A by a dense open, we may assume that the inertia group scheme I x A of x A ∈ X (A) is finiteétale over A. (The existence of such a dense open follows from spreading out of finiteétale morphisms.) Write B = Spec A.
Let y in X (A) with y k ∼ = x k in X (k). Then there exists a finite field extension L of K such that the B-scheme Isom B (x, y) has an L-point. Since the diagonal of X is proper, the morphism Isom B (x, y) → B is proper. In particular, since Isom B (x, y) → B is proper and has an L-point, as B is normal it follows that the latter has a C-valued point, where C → B is the normalization of B in L. This shows that the morphism Isom B (x, y) → B is an I x -torsor. The set of y in X (A) with y k ∼ = x k in X (k) is therefore a subset of H 1 (B, I x ). Since B is an arithmetic scheme and I x → B is a finiteétale group scheme, the latter set is finite by Lemma 3.5. We conclude that (1) =⇒ (2).
We now show that (2) implies (3). Let X be as in (3). As X is separated, its diagonal is proper (by definition). Therefore, since the diagonal of X over A is proper and affine, it is finite. We claim that this implies that the map of sets π 0 (X (A)) → π 0 (X (K)) is injective. Indeed, for all x, y ∈ X (A), by Zariski's Main Theorem, a generic section of the finite morphism Isom A (x, y) → Spec A induces a section (as Spec A is integral and normal). In other words, if x K is isomorphic to y K in X (K), then x is isomorphic to y in X (A). This concludes the proof of (2) =⇒ (3).
We now show that (3) implies (1). To do so, let A ⊂ k be a Z-finitely generated subring A ⊂ k and let X be a model for X over A such that X is a separated algebraic stack with affine diagonal. (The existence of such a model follows from the fact that separatedness and "having affine diagonal" spread out; see [44,Prop. B.3].) Let A ⊂ A ′ ⊂ k be a smooth Z-finitely generated subring containing A. Since separatedness and "having affine diagonal" are stable by base-change, the algebraic stack X A ′ is separated and has affine diagonal over A ′ . Therefore, by (3), the set π 0 (X (A ′ )) is finite. In particular, the set Im[π 0 (X (A ′ )) → π 0 (X (k))] is finite. We conclude that X is arithmetically hyperbolic over k from Lemma 4.9.
Remark 4.19. The conclusion of Theorem 4.18 can fail for separated non-Deligne-Mumford algebraic stacks over k. Indeed, by [35, p. 241], there is an integer n ≥ 1 and infinitely many Q-isomorphism classes of smooth proper genus 1 curves C over Q with good reduction outside n. Therefore, by Shafarevich's finiteness theorem for elliptic curves [47, Thm. IX.6.1], there is an elliptic curve E over A := Z[1/n] with infinitely many pairwise non-isomorphic twists (over A); thus is infinite. In particular part (2) of Theorem 4.18 does not hold in this case, even though BE is arithmetically hyperbolic for any elliptic curve E over k (Lemma 4.14). (To see that BE → Spec k is separated, note that E = Spec k × BE Spec k is proper over k.) This also shows that the smooth arithmetically hyperbolic finitely presented separated stack M 1 of smooth proper genus one curves fails Theorem 4.18. (For S a scheme, every object of M 1 (S) is a smooth proper morphism f : X → S of algebraic spaces whose geometric fibres are smooth proper connected curves of genus one.) Remark 4.20. Let G be an affine finite type group scheme over Z. Note that the finitely presented algebraic stack BG k is arithmetically hyperbolic over k for "trivial" reasons (Lemma 4.14). We expect that the integral points on BG satisfy a stronger finiteness property. Indeed, it seems reasonable to suspect that, for all finitely generated subrings A ⊂ k with K := Frac(A), the set is finite; see [8,9,18,24] for related results. For instance, by [18,Prop. 5.1], this finiteness holds if dim A = 1. Also, in [8,Thm. 8.1.(i)], this finiteness result is proven under the assumption that dim A = 2 and G is an affine finite type group scheme such that G K is a simple algebraic group over K of G 2 -type (and also in some other cases).
Remark 4.21. Many natural moduli problems over Q are finitely presented separated Deligne-Mumford stacks over Q. However, the natural model for such a stack over Z might not be Deligne-Mumford nor separated. For instance, the stack of smooth plane cubic curves C (3;1) is a finite type separated algebraic stack with affine diagonal which is Deligne-Mumford over Z[1/3], but not over Z.

Chevalley-Weil for algebraic stacks
The classical theorem of Chevalley-Weil says that, if f : X → Y is a finiteétale morphism of algebraic varieties over Q, then X is arithmetically hyperbolic if and only if Y is arithmetically hyperbolic. This theorem can be considered as an arithmetic analogue of the similar statement for Brody hyperbolic varieties: if X → Y is a finité etale morphism of algebraic varieties over C, then X is Brody hyperbolic if and only if Y is Brody hyperbolic.
We generalize this to properétale morphisms of algebraic stacks which may not be representable (e.g. gerbes), using our version of Hermite-Minkowski.
Theorem 5.1. Let f : X → Y be a properétale morphism of finitely presented algebraic stacks over an algebraically closed field k of characteristic 0. Then X is arithmetically hyperbolic if and only if Y is arithmetically hyperbolic.
Proof. Note that X → Y is quasi-finite. In particular, if Y is arithmetically hyperbolic, then it follows from Proposition 4.16 that X is arithmetically hyperbolic.
So assume that X is arithmetically hyperbolic. Let A ⊂ k be a smooth finitely generated Z-algebra and let X → Y be a properétale morphism of finitely presented algebraic stacks over A which is isomorphic to X → Y after base-change to k and for which the degree of X → Y is invertible on A (in the sense defined right before Theorem 3.9); such data exists because properétale morphisms spread out [44,Prop. B.3]. By Lemma 4.9 it suffices to show that Im[π 0 (Y(A)) → π 0 (Y(k))] is finite.
If Spec A → Y is an A-point, then we have a Cartesian diagram In particular, as X and Y are fixed, the degree and inertia degree are also fixed. Thus by Hermite-Minkowski for stacks (Theorem 3.9), the set of A-isomorphism classes of algebraic stacks B appearing as the pull-back of an A-point of Y is finite. Thus, we can simultaneously "trivialize" every B appearing from this construction. That is, there is a Z-finitely generated subring A ′ ⊂ k containing A such that, for all B appearing from the above construction, there is a morphism Spec A ′ → B. We fix for each B such a morphism Spec A ′ → B. In a diagram, the situation looks as follows: ). Now, since Y C is arithmetically hyperbolic over C and Y A ′ is a finite type separated Deligne-Mumford algebraic stack over A ′ , it follows from the twisting lemma (Theorem 4.18) that the groupoid Y (A) is finite.

Applications
We now give applications of our results to certain surfaces of general type and prove a transcendental criterion for arithmetic hyperbolicity.
6.1. Application to the moduli of surfaces of general type. Let p and q be integers. Let S p,q be the stack of canonically polarized surfaces X with p g (X) = p and q(X) = q. Thus, for S a scheme, the objects of the groupoid S p,q (S) are smooth proper morphisms X → S of schemes whose geometric fibres X s are smooth projective connected (minimal) surfaces with ample canonical bundle such that p g (X s ) = p and q(X s ) = q. By Matsusaka-Mumford [34], the stack S p,q is a locally finite type separated algebraic stack over Z. Moreover, "boundedness" for canonically polarised surfaces with fixed p g and q implies that S p,q is of finite type over Z; see [30,Thm. 1.7]. Finally, as S p,q parametrizes proper varieties with a (canonical) polarization, the diagonal of S p,q is affine (cf. the proofs of [ It seems reasonable to suspect that S p,q,C is arithmetically hyperbolic (cf. [22, Conjecture 1.1]). Indeed, its subvarieties are of log-general type by a theorem of Campana-Paun [7]. Moreover, its subvarieties are Brody hyperbolic [51] and even Kobayashi hyperbolic [49,45]. Also, the stack S p,q,C satisfies the "function field" analogue of arithmetic hyperbolicity by a theorem of Kovács-Lieblich [31]. Thus, in light of Lang-Vojta's conjecture, ignoring stacky issues, it seems reasonable to suspect that S p,q,C is arithmetically hyperbolic over C. Our next result gives a modest contribution towards this expectation, and illustrates how one can use our results by arguing directly on the moduli stack. Theorem 6.1. For q ≥ 4, the stack S 2q−4,q,C is arithmetically hyperbolic over C.
Proof. For g ≥ 2, let M g be the stack of smooth proper curves of genus g over Z.
Let f : M 2,C × M q−2,C → S 2q−4,q,C be the morphism which associates to a pair (X, Y ) in M 2,C × M q−2,C the object X × Y in S 2q−4,q,C . Note that f is a welldefined morphism of algebraic stacks over C. This morphism is surjective by the classification of minimal surfaces of general type X with p g (X) = 2q(X) − 4 over the complex numbers (see Beauville's theorem in the appendix to [13]). In particular, since M 2,C × M q−2,C is connected, the algebraic stack S 2q−4,q,C is connected. Note that, by [5, Thm. 1.1] and the connectedness of S 2q−4,q,C , this morphism is finiteétale. For every g ≥ 2, the stack M g,C is arithmetically hyperbolic over C (Example 4.6), so that the stack M 2,C × M q−2,C is arithmetically hyperbolic over C (Lemma 4.11). Therefore, as f : M 2,C × M q−2,C → S 2q−4,q,C is finiteétale, the result follows from the Chevalley-Weil theorem (Theorem 5.1).
Proof of Theorem 1.3. Since S 2q−4,q is a finite type separated algebraic stack with affine diagonal over Z and S 2q−4,q,C is arithmetically hyperbolic over C (Theorem 6.1), it follows from the twisting lemma (Theorem 4.18) that π 0 (S 2q−4,q (A)) is finite.

A transcendental criterion.
In this section we use Faltings's finiteness theorem (formerly the Shafarevich conjecture for principally polarized abelian varieties) and Borel's algebraization theorem to prove a "transcendental" criterion for arithmetic hyperbolicity (Theorem 6.4). We view this criterion as a confirmation of Lang's philosophy that complex analytic hyperbolicity should have arithmetic consequences.
For an integer g, let A g be the stack over Z of g-dimensional principally polarized abelian schemes. We recall some properties of A g proven, for instance, in [37,39]. The stack A g is a smooth finite type separated Deligne-Mumford algebraic stack over Z whose coarse space is a quasi-projective scheme over Z. For n ≥ 1, let A [n] g be the stack over Z[1/n] of g-dimensional principally polarized abelian schemes with a full level n-structure. Note that the forgetful functor A Our goal in this section is provide a precise interplay between the "analytic" hyperbolicity of A g,C (i.e., A an g,C is hyperbolically embedded in its Baily-Borel compactification) and the arithmetic hyperbolicity of A g,C (as proven by Faltings). We start with an analytic property of A an g,C (which is also studied in [23]). Lemma 6.2 (Borel's algebraization theorem). Let X be a finite type reduced scheme over C, and let ϕ : X an → A [3],an g,C be a morphism. Then ϕ is algebraic, i.e., there is a unique morphism of schemes f : X → A [3] g,C such that f an = ϕ. Proof. The uniqueness of f is clear. Since A [3],an g,C is a locally symmetric variety, the result follows from Borel's theorem [6, Thm. 3.10] (see also [14,Thm. 5

.1]).
We now prove a generalization of Borel's algebraization theorem to stacks. To state this result, for X a finitely presented algebraic stack over C, we let X an be the associated complex-analytic stack; see [41, §6.1] for a definition of the stack X an . Proposition 6.3 (Stacky Borel algebraization). Let X be a finitely presented reduced algebraic stack over C, and let ϕ : X an → A [3],an g,C be a morphism. Then ϕ is algebraic, i.e., there is a unique morphism of stacks f : X → A [3] g,C such that f an = ϕ. Proof. As A [3] g,C is a scheme, the functor Hom(·, A [3] g,C ) is a sheaf for the fppf topology on stacks over C. Let P → X be a smooth surjective morphism with P a finite type reduced scheme over C. Then, by Borel's algebraization theorem (Lemma 6.2), the morphism P an → A [3],an g,C is the analytification of a unique morphism P → A [3] g,C . Similarly, the morphism (P × X P ) an = P an × X an P an → A [3],an g,C is the analytification of a unique morphism P × X P → A [3] g,C . Thus, by the sheaf property of Hom(·, A [3] g,C ), we conclude that the morphism ϕ : X an → A [3],an g,C is the analytification of a unique morphism f : X → A [3] g,C . Theorem 6.4 (Transcendental criterion). Let X be a finitely presented algebraic stack over C. If there exist a finitely presented algebraic stack Y , a properétale morphism Y → X and a quasi-finite holomorphic map Y an → A an g,C , then X is arithmetically hyperbolic over C.
Proof. We may and do assume that X is reduced (Lemma 4.15). In particular, Y is a reduced finitely presented algebraic stack over C. Moreover, let Y ′ = Y an × A an g,C A [3],an g,C . Then the natural holomorphic map Y ′ → Y an is finiteétale. Therefore, by the stacky version of Riemann's existence theorem [38,Thm. 20.4], there is a reduced finitely presented algebraic stack Z over C, a finiteétale morphism Z → Y , and an isomorphism Z an ∼ = Y ′ over Y an . Note that Z an → A [3],an g,C is a quasi-finite holomorphic map. By stacky Borel algebraization (Proposition 6.3), there is a (quasifinite) morphism Z → A [3] g,C . By Faltings's finiteness theorem, the stack A g,C is arithmetically hyperbolic over C; see [16] (which builds on [15]). Therefore, the scheme A [3] g,C is arithmetically hyperbolic by Proposition 4.16. Since A [3] g,C is arithmetically hyperbolic over C, it follows from Proposition 4.16 that Z is arithmetically hyperbolic over C. By the Chevalley-Weil theorem (Theorem 5.1), as Z → Y → X is properétale, we conclude that X is arithmetically hyperbolic over C.
6.3. Application to cubic threefolds. Being able to pass to a properétale cover in Theorem 6.4 is very natural for applications. Such covers of moduli stacks often naturally arise by adding level structure to the objects parametrized by the stack, where Y is usually even a scheme. In practice, morphisms to A an g,C arise via period maps, and such a morphism being quasi-finite translates to a (local) Torelli theorem.
We give an application of this type, which is a new proof of the arithmetic hyperbolicity of the moduli of smooth cubic threefolds [25,Thm. 1.1]. This proof is much simpler than the proof given in [25], as it avoids the need to create an algebraic theory of intermediate Jacobians, i.e. it avoids the need to construct the intermediate jacobian C (3;3),C → A 5,C as a morphism of algebraic stacks over C (hence also avoids the need for an arithmetic theory of intermediate Jacobians, given by a morphism of stacks C (3;3),Q → A 5,Q ).
Proof. Let X := C (3;3),C . Since X is uniformisable by an affine scheme [26], there is an affine variety Y over C and a finiteétale morphism Y → X. Let V be the polarized variation of Hodge structures on Y associated to the pull-back of the universal family U → X along Y → X. Let Y an → A an 5,C be the associated period map. By infinitesimal Torelli for smooth cubic threefolds, the latter morphism is injective on tangent spaces. In particular, it has finite fibres (see for instance [25,Thm. 2.8]). Therefore, the arithmetic hyperbolicity of X over C follows from the transcendental criterion (Theorem 6.4).