Constructing abelian varieties from rank 2 Galois representations

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field $E\subset \bar{\mathbb Q}_\ell$, and has bad, infinite reduction at some closed point $x$ of $X\setminus U$. We show that $\mathbb L$ occurs as a summand of the cohomology of a family of abelian varieties over $U$. The argument follows the structure of the proof of a recent theorem of Snowden-Tsimerman, who show that when $E=\mathbb Q$, then $\mathbb L$ is isomorphic to the cohomology of an elliptic curve $E_U\rightarrow U$.


Introduction
To state our main result, we require the following definition and setup.Definition 1.1.Let B/k be a smooth variety over a finitely generated field and let ℓ = char(k) be a prime.An abelian scheme g : A B → B is said to be of SL 2 -type if there is a decomposition of lisse Q ℓ -sheaves on B: where each L i is a geometrically irreducible rank 2 lisse Q ℓ -sheaf on B with cyclotomic determinant: Setup 1.2.Let K be a number field and let X/K be a smooth, proper, geometrically irreducible curve.Let U ⊂ X be a Zariski open and dense subset of X with reduced complementary divisor D. Assume that D is non-empty.
Let f : A U → U be a generically simple abelian scheme that is of SL 2 -type and has bad, infinite reduction along some non-empty subset of D. Then the following statements hold for each direct summand Theorem 1.3 (Snowden-Tsimerman).Notation as in Setup 1.2 and let L be a lisse Q ℓsheaf on U satisfying the above conditions (1)- (4), with E = Q.Then there exists a family of elliptic curves f : In this article, we consider the situation where Frobenius traces are all contained in a fixed number field E.
An observation of Litt implies that for an arithmetic local system, (2) will automatically, hold: see Step 2 of the proof of [17,Theorem 1.1.3]or [22,Theorem 6.1].(See also the argument in [18,Proposition 4.1].)Therefore, to prove the relative Fontaine-Mazur conjecture for rank 2 local systems that have infinite monodromy around some point, it suffices to bound the field generated by Frobenius traces.This task seems to be quite difficult in general; for some progress on this question, see [24].
Remark 1.6.We do not have any idea how to get around point (4).As will be explained in the proof sketch, this is because we crucially use some of Drinfeld's early work on the Langlands correspondence for GL 2 over function fields.More specifically, he is able to show that if L is an irreducible rank 2 lisse Q ℓ -sheaf over a curve U/F q with cyclotomic determinant and infinite monodromy at ∞, then L comes from a family of abelian varieties over U. His proof finds such an abelian scheme as an isogeny factor of a Drinfeld modular curve over F q (U). 1 When we do not assume infinite monodromy at ∞, then no such result is known; more specifically, the output of his later work on the Langlands correspondence will imply that there exists an open subset V ⊂ U ×U and a smooth projective morphism f : S → V of relative dimension 2 such that L ⊠ L * | V is a summand of R 2 f * Q ℓ .See [12, Remark 1.4, Question 9.1] and [13, Section 1] for related discussion.
Our argument largely follows [25], but we need several new ingredients.To explain this, we quickly reprise their argument in the following remark.
(1) Using Drinfeld's first work on the Langlands correspondence over finite fields, for for all p ≫ 0 they construct families of elliptic curves over U p with trivial ℓ 3 torsion whose monodromy is isomorphic to ρ| Up .(This involves an implicit isogeny from what is produced by Drinfeld's theorem.)These families in turn induce maps where M1,1 (ℓ 3 ) is the compactified moduli space of elliptic curve with full ℓ 3 level structure, defined over Spec(Z[1/ℓ]), and the final target is therefore a hyperbolic curve over O K /p.(2) While the map λ p is not a priori generically separable, they factor it through absolute Frobenius to construct a new map, µ p : X p → M1,1 (ℓ 3 ) ⊗ O K /p which is generically separable.Note that the induced elliptic curve over U p also has monodromy isomorphic to ρ| Up .Then Riemann-Hurwitz applies, bounding the degree of the map µ p by some number d, which is crucially independent of p.We may replace λ p with µ p .(3) At this point, consider the moduli space of maps: )), of morphisms of curves λ over O K [1/N], with degree bounded by d.This moduli space is a scheme of finite type over O K [1/N] because we have put a bound on the degree. 2For each k, let H k denote the subset of H consisting of those maps λ such that: • λ(U) ⊂ M 1,1 (ℓ 3 ); • and the induced elliptic curve E U → U has mod ℓ k monodromy isomorphic to ρ mod ℓ k .Then Snowden-Tsimerman show that each the H k is a closed subset, and hence so is H ∞ := ∩H k .The subset H ∞ ⊂ H will parametrize those maps λ such that the monodromy representation is integrally isomorphic to ρ. Equipping H ∞ ⊂ H with the reduced induced subscheme structure, they deduce that H ∞ is therefore a scheme of finite type over O K [1/N].As it has points modulo p for infinitely many primes p of O K , it follows that it has a point over a finite extension field K ′ /K.Then a Weil restriction argument, together with Faltings' isogeny theorem, allows one to conclude.
We now explain the new ingredients in turn, highlighting the additional difficulties.
Remark 1.8 (Sketch of the proof of Theorem 1.4).Again for notational simplicity, assume that L corresponds to a representation: (Note that Q ℓ contains number fields of infinitely large degree!)We further assume that ρ has the property that the mod ℓ 3 residual representation π 1 (U K ) → GL 2 (Z/ℓ 3 Z) is trivial.(This last assumption will play no role, but we include it to see which additional technicalities emerge.) (1) Again using Drinfeld's early work on the Langlands correspondence over finite fields, for each p ≫ 0, we may construct an abelian scheme over f p : (a) Here we encounter our first complication: it is not necessarily true that we can choose A p [ℓ 3 ] to be the split étale cover of U p : unlike in the case [25], L| Up is not all of R 1 f p, * Q ℓ .However, there exists a finite, connected cover The key property of the cover ϕ : (X p ) ′ → X p is that the degree is bounded independent of p.However, we emphasize that, as of yet, there is no preferred X ′ → X over K that patches all of these modulo p covers together.At this point, we demand that N > |GL 2h (Z/ℓ 3 Z)|, to ensure that any such ϕ p is tamely ramified.(b) We now encounter our next (minor) trouble.A priori, there is no bound on the degree of the polarization of f ′ p : A ′ p → (U p ) ′ .This has a simple solution: Zarhin's trick, which says that B ′ p := (A ′ p × (A ′ p ) t ) 4 has a principal polarization.(c) There is a third trouble; unlike in the approach of Snowden-Tsimerman, we have not yet nailed down the integral monodromy, and this is more subtle.
There are several ways one could address this.Our solution to this problem will be found in the construction of a simple moduli space, H k : see Step (3).We therefore get a map: 8h,1,ℓ 3 is the Baily-Borel compactification of the fine moduli scheme A 8h,1,ℓ 3 parametrizing principally polarized abelian schemes of dimension 8h and trivial level ℓ 3 structure.This λ ′ p has the following property: the pullback of the universal rank 16h lisse ℓ-adic sheaf on A 8h,1,ℓ 3 to U p has ρ as a rational summand.
(2) Our next goal is to somehow numerically bound λ ′ p .Recall that [25] do this by a combination of Riemann-Hurwitz and factoring through some power of absolute Frobenius.In our setting, this step is more tricky, and we chose to use an Arakelov-style inequality.More precisely, if f p : B′ p → (X p ) ′ is the Néron model of B ′ p → U ′ p , then we will bound the degree of the Hodge vector bundle E 1,0 (Xp ′ (log ∆), at least for many infinitely many p.Set E 0,1 (Xp) Then to bound the degree of E 1,0 (Xp) ′ , we will need to know that the logarithmic Kodaira-Spencer map constructed by Faltings-Chai, is not only nonzero but is moreover an isomorphism at the generic point. 3In more detail: For any p ≫ 0 such that the underlying prime number p splits completely in E, the field generated by Frobenius traces, the induced p-divisible group on (U p ) ′ splits as the direct sum of several copies of h (mutually non-isogenous) height 2, dimension 1 p-divisible groups G ′ i and their duals (G ′ i ) t .We prove, using monodromy considerations, that they are generically ordinary and have supersingular points.Applying a Frobenius untwisting lemma from the PhD thesis of Jie Xia [28], we conclude that we may "Frobenius untwist" each of them until they are all generically versally deformed. 4(In the appendix, we provide a proof of Xia's Frobenius untwisting lemma in our context, and also give a second argument and perspective of the termination of Frobenius untwisting stability techniques.)Once again using Zarhin's trick, we will obtain an isogenous, principally polarized abelian scheme over U ′ p , which we relabel B ′ p , with the Néron model and such that the logarithmic Kodaira-Spencer map is a generically injective map of coherent sheaves on (X p ) ′ .By taking determinants, we deduce an Arakelovstyle inequality, thereby bounding the degree of the induced Hodge line bundle on (X p ) ′ by some integer d, which is crucially independent of p.The output of this is Lemma 2.7.(3) To mimic the third step, we first construct some finite type moduli spaces of O K [1/N], and then we use our argument as above to show it has points modulo p for infinitely many p.In greater detail: (a) Fix d > 1 and set H to be the moduli of triples (X ′ , ϕ, λ): where is a smooth, proper, geometrically connected curve; • ϕ is finite, of degree at most ≤ |GL 16h (Z/ℓ 3 Z)|, and étale over U; • there exists some point ∞ ′ ∈ X ′ that is sent to a 0-dimensional cusp in A * 8h,1,ℓ 3 ; • and the degree of the pulled-back Hodge line bundle on X ′ K is ≤ d.Then H will be a Deligne-Mumford stack, of finite type over O K [1/N].(The stackiness comes from the intervening Hurwitz space.)We further show that the generic fiber of H has dimension 0 (and is in fact reduced); while this is an immediate corollary of the PhD thesis of Ben Moonen, we argue following the work of Saito.This implies that, after potentially further increasing N, the stack H/O K [1/N] has relative dimension 0.5 (b) Recall that L is a lisse Z ℓ -sheaf on U, whose generic fiber is an Z ℓ -lattice inside of L. For k ≥ 1, set H k to be the subspace of H given by those (X ′ , ϕ, λ) (with induced abelian scheme f : B ′ → X ′ ) such that there exists a map ψ : ϕ * (L)/ℓ k → R 1 f * Z ℓ /ℓ k of torsion locally constant étale sheaves with the following condition: the reduction modulo ℓ of ψ is nonzero.(This condition is crucial in our approach.) 6Then H k ⊂ H will be a closed substack, which we may equip with the reduced induced structure.Similarly, set H ∞ to be ∩H k , again with the reduced induced stack structure.Note that H ∞ is then a finite type Deligne-Mumford stack over Spec(O K [1/N]) for some N. Unlike in the Snowden-Tsimerman approach, the relationship of the moduli space H ∞ to Drinfeld's theorem is not immediately apparent.However, in both approaches, the moduli spaces involve extra maps of ℓ k -torsion sheaves rather than lisse Z ℓ -sheaves.(c) By the careful choice of H k and a crucial diagonalization argument on H ∞ (contained in Lemma 3.6), it will follow from the earlier steps that there exists an infinite set of primes p such that H ∞ has points modulo p. (Unlike the approach of Snowden-Tsimerman, this does not require one take an ℓ-primary isogeny.)By the Nullstellensatz, one deduces that H ∞ has characteristic 0 points.A Weil restriction argument then yields the result.
Remark 1.9.Katz has shown that rigid local systems on the punctured projective line are motivic, and Corlette-Simpson have shown that all rigid rank 2 local systems are motivic.Our main theorem provides a new arithmetic approach to both Katz's theorem in rank 2 and also the Corlette-Simpson theorem, subject to an additional assumption analogous to (4).Here is an outline of the proof.We emphasize that these approaches will critically rely on a quasi-projective version of a deep theorem on projective varieties of Esnault-Groechenig [5], which is not yet available.First we assume that U is a curve.Let L be a cohomologically rigid local system of rank 2 on U an C with coefficients in Q ℓ , trivial determinant and infinite monodromy around ∞. Suppose that the local system L spreads out to an étale local system L with cyclotomic determinant over a finitely generated spreading out U/S such that the stable Frobenius trace fields are bounded, i.e., there exists a number field E such that for all closed points s, there exists a finite extension s ′ /s such that the Frobenius trace field of L| U s ′ is contained in E.Then, our argument applies verbatim to prove that L over U C comes from a family of abelian varieties; we get mod p points for infinitely many p ≫ 0, and the relevant moduli space is of finite type and in fact generically 0 dimensional, so by specialization of the prime-to-p fundamental group we may conclude.
In fact, recent work of the first-named author and J. Lam shows the following.If X/C is a projective variety, and if L is a cohomologically rigid Q ℓ -local system with trivial determinant on X an , then there exists a spreading out X/S and a number field E ⊂ Q ℓ such that L spreads out to an étale local system L on X with cyclotomic determinant such that the stable Frobenius trace field is contained inside of E. In the quasi-projective case, which is what is needed here, the arguments of loc.cit.should go through once a quasi-projective analog of the theorem of [5] is available: we need that cohomologically rigid stable flat connections give rise to F f -isocrystals on the relevant p-adic completions.
In general, when U is higher dimensional (i.e., U = X \ D, where X is a smooth projective variety and D is a simple normal crossings divisor), it is very plausible that once the quasi-projective analog of [5] is available, one may similarly deduce the analog of the Corlette-Simpson theorem here (again, subject to the restriction that the local monodromy around one of the boundary divisors is infinite).Here is a sketch of the argument.The putative quasi-projective version of [5] will in fact output rank 2 filtered logarithmic F -crystals; as above, porting these objects into the techniques of Krishnamoorthy-Lam, one can deduce that cohomologically rigid rank 2 local systems will have spreading-outs whose stable Frobenius trace field is bounded.A complete set of companions to the logarithmic F -isocrystals so constructed will likely exist, as in the projective case this is shown in [5].From these F -isocrystals, [14] will provide abelian schemes on open subsets of the mod p fibers of bounded dimension7 and [13, Corollary 6.12] shows that, after possibly replacing with an isogenous abelian scheme, the abelian schemes extend to the whole mod p fiber of U. We can bound the degree of the Hodge line bundle for infinitely many p by Frobenius untwisting, exactly as in done here.Finally, the appropriate Hom scheme will again be 0-dimensional, so by using specialization of the prime-to-p fundamental group one may again conclude.

Drinfeld's work on the Langlands correspondence for GL 2 and some corollaries
A key ingredient in the proof of Theorem 1.4 is the following Theorem 2.2, which is a byproduct of Drinfeld's first work on the Langlands correspondence for GL 2 .We first record a setup.Setup 2.1.Let p be a prime number and let q = p a .Let C/F q be a smooth, affine, geometrically irreducible curve with smooth compactification C. Let Z := C \ C be the reduced complementary divisor.Theorem 2.2.(Drinfeld) Notation as in Setup 2.1 and let L be a rank 2 irreducible Q ℓ sheaf on C with determinant Q ℓ (1).Suppose L has infinite local monodromy around some point at ∞ ∈ Z. Then L comes from a family of abelian varieties in the following sense: let E be the field generated by the Frobenius traces of L and suppose [E : Q] = h.Then there exists an abelian scheme We make some observations about the p-adic properties of the resulting abelian schemes.In particular, our goal is to show that, in the context of Theorem 2.2, we can modify A C → C with products, duals, and isogenies such that the resulting abelian scheme B C → C that has especially nice (p-adic) properties; these will in turn allow us to prove an Arakelov-style inequality.First, we will give the following non-standard definition, which is adapted for our purpose.
Definition 2.3 is useful as it concisely expresses the condition that G C have semistable reduction and moreover that it does not extend as a p-divisible group across any of the cusps.
The next proposition will be critical for bounding degrees of maps to moduli spaces.In the appendix, we explain a second proof/perspective of the second part, which is based on a destabilizing iteration argument due to Langer.Before we begin the proof, we comment on the overconvergence assumption.If H C → C is a p-divisible group, then F -isocrystal D(H C ) is automatically a convergent F -isocrystal.In our setting, the fact that we demand G C → C to be semistable around Z implies that D(G C ) is in fact overconvergent.Part of the hypothesis of Proposition 2.4 is then that First proof of Proposition 2.4.As G C has height 2 and dimension 1, there are only two possible Newton polygons, which correspond to the p-divisible group being ordinary or supersingular respectively.If G C were not generically ordinary, it would be everywhere supersingular.However, supersingular p-divisible groups cannot be strictly semistable: as there is no multiplicative part, the filtration in [2, Definition 2.2], would have to be trivial, which would imply that G C extends to a p-divisible group over C.This shows G C is generically ordinary.
Suppose that G C had no supersingular points: then G C is everywhere ordinary.Let H C be the multiplicative sub p-divisible group of G C , i.e., the height 1, dimension 1 p-divisible group with Newton slope 1 everywhere.Let ∞ ∈ Z, with formal parameter z ∞ .Then the p-divisible group G C | Spec(Fq((z∞))) has semistable reduction in the sense of 8 We briefly recall the notion here.Let G C → C be a height 2, dimension 1 p-divisible group.There is a Kodaira-Spencer map: KS : T C → Ω * ⊗ Ψ, where Ω is the Hodge line bundle of G C and Ψ is the dual of the Hodge line bundle of the Serre dual G t C .We say that G C → C is generically versally deformed if the above KS is nonzero.After the work of Illusie [8], this is equivalent to the following condition: there exists a closed point c such that the map u c : C ∧ c → Def(G c ) from the formal completion of C at c to the equal-characteristic universal deformation space of G c is a formally smooth map of formally smooth, 1-dimensional κ(c) schemes, i.e., u c is an isomorphism.
[2, Definition 2.2] and does not extend to a p-divisible group over Spec(F q [[z ∞ ]]).Then, for the definition of semi-stability to be satisfied, the only possible filtration is: We claim this process must terminate at some point.Here is a simple proof (also indicated in [12, p. 253]).Let c be a closed point of C such that G c is supersingular.Then the (equal characteristic) deformation map: ] is nonzero, because G C is generically ordinary.(In other words, if u c were 0, then the p-divisible group over Spec(κ(c)[[z c ]]) would be supersingular at both the closed and the generic point, which is a contradiction: over the generic point, the p-divisible group is base-changed from F q (C) along the map The map KS c is simply the derivative of u c .In particular, KS c = 0 implies that u * c (t) ∈ κ(c)[[z c ]] is a power series in z p c ; on the level of the universal deformation map u c , Frobenius untwisting amounts to extracting a p th root of u * c (t).As u c is not constant, this process must terminate.
• The lisse Q ℓ -sheaf L has infinite, unipotent local monodromy around each point ∞ ∈ Z. • Let E be the field generated by Frobenius traces of L. Suppose that p splits completely in E. Then there exists an abelian scheme f C : A C → C satisfying all of the conclusions of Drinfeld's Theorem 2.2, together with the following further properties. where (1) the G C,i are all mutually non-isogenous; (2) each G C,i is a height 2, dimension 1 p-divisible group on C; and (3) each G C,i is generically versally deformed, generically ordinary, and has nonempty supersingular locus.
Proof.We will first construct i G C,i with the desired properties.
Let f : A C → C be an abelian scheme produced by Drinfeld's Theorem 2.2.By Grothendieck's monodromy criterion for semistable reduction, A C → C is totally degenerate around every point of Z.The F -isocrystal [21].We claim that E is the companion to R 1 (π C ) * Q ℓ .Indeed, a theorem of Zarhin [20, Chapitre XII, Theorem 2.5, p. 244-245] implies that R 1 (π C ) * Q ℓ is semi-simple and the characteristic polynomials of Frobenius agree at closed points by [11].On the other hand, there is a decomposition: (2.1) where E i are irreducible objects of F-Isoc † (C) Q p It follows from [14, Remark 2.8] that every summand E i is a companion of L. 9 As the relation of companions preserves "infinite monodromy at ∞ ∈ Z", each E i has infinite monodromy around every ∞ ∈ Z. Also, det(E i ) = Q p (1), again because the property "cyclotomic determinant" is preserved under the companions relation.
As p splits completely in E, it follows that E ⊗ Q p ∼ = ΠQ p acts on E, and the images of the idempotents are the E i , i.e., the (absolutely) irreducible summands E i are objects of F-Isoc † (C).
The slopes of each E i are in between 0 and 1. Therefore we may apply [13, Lemma 5.8]10 and [1] to see that for each E i , there exists a (non-canonical) , each E i has coefficients in Q p , by the hypothesis that p splits completely in E.) The p-divisible groups A C [p ∞ ] and i G C,i are isogenous.At this point, we wish to claim that each G C,i has strong strict semistable reduction along Z.First of all, note that A C [p ∞ ] has has strong strict semistable reduction by [2, 2.5].
As D(G C,i ) ⊗ Q p is overconvergent, it follows from [21, Theorem 2.22] that every G C,i has semistable reduction along Z. Suppose for contradiction that G C,1 extended through some cusp ∞ ∈ Z. Then E 1 ∼ = D(G C,1 ) ⊗ Q p also extends to an (overconvergent) Fisocrystal on the curve C ∪ {∞} = C \ {Z \ ∞}.As each of the E i are companions, this implies that they all also extend to C ∪ {∞}.Therefore the ℓ-adic companion also extends to C ∪ {∞}.This implies that L also extends to a lisse Q ℓ -sheaf on C ∪ {∞}, contradicting our assumption that L had infinite, unipotent monodromy around ∞.
We may now apply Proposition 2.4 to replace each G C,i with an isogenous p-divisible group that satisfies the two conclusions of the proposition.Note that we still have the relation: . By [14, Lemma 2.13], it follows that we can replace A C by an isogenous abelian scheme such that: where every G C,i is generically versally deformed, is generically ordinary, and has supersingular points.Finally, each G i will be mutually non-isogenous because the F -isocrystals D(G i ) ⊗ Q p are a complete collection of p-adic companions of L [14, Remark 2.8].
Using the above, we will be able to extract all of the p-adic information we need from Theorem 2.2 to prove Theorem 1.4.We need one final piece of notation.Definition 2.6.Let N ≥ 1 be an integer prime to p and let g ≥ 1 be a positive integer.Then A g,1,N /Spec(Z[1/N]) denotes the (fine) moduli space of principally polarized abelian varieties with trivial full level N structure.This is a smooth, quasi-projective scheme over Spec(Z[1/N]).It has a compactification, A * g,1,N /Spec(Z[1/N]). 11This latter scheme has a natural ample line bundle, the Hodge line bundle, which we denote by α.
Then the precise output we need from Drinfeld's Theorem 2.2 is given in the following lemma.
Lemma 2.7.Notation as in Theorem 2.2.Suppose that p splits completely in E and L has infinite, unipotent monodromy around every point of Z.
Then there exists a principally polarized abelian scheme abelian scheme f : B C → C, of SL 2 -type and dimension 8h, such that L occurs as a direct summand of R 1 f * Q ℓ , and the following holds.
(1) B C → C has semistable, infinite reduction along C\C.Call the Néron model ) There exists h mutually non-isogenous p-divisible groups G C,i , each of height 2 and dimension 1 and generically versally deformed, such that there is a decomposition of p-divisible groups (3) After Kato-Trihan, to f : B C → C there is an associated logarithmic F -crystal with nilpotent residues (M, F ) in finite, locally free modules on the log pair ( C, Z).
Similarly, there is a logarithmic Hodge vector bundle, which we write Ω B/ C , a rank 8h vector bundle on C.Then the following hold.
(i) Ω B/ C splits as the direct sum of 8h positive line bundles on C; (ii) the log Kodaira-Spencer map (constructed in [7, Ch.III, Cor 9.8]: where ω C denotes the sheaf of differential one-forms on C, is an injective map of coherent sheaves on C; Then the induced moduli map C → A 8h,1,N extends to a map: , where the first transpose is "dual abelian scheme" and the second is "Serre-dual p-divisible group", it follows that (2) holds.
We are left to prove (3).To do this, we will heavily use [14,Setup A.10,Proposition A.11].First of all, each D(G C,i ), a priori an Dieudonné crystal on C, extends uniquely to logarithmic Dieudonné crystal (with nilpotent residues) on ( C, Z).Indeed, existence of the extension of D(B C [p ∞ ]) follows from [9, 4.4-4.8]and uniqueness from [14,Proposition A.11(3)]: name the extension (M, F, V ).These two results immediately imply the desired existence and uniqueness for the extension of D(G i ) to a logarithmic F -crystal, which we name (M i , F, V ).The uniqueness implies that the (unique) extension of D(G t i ) is isomorphic to the dual logarithmic Dieudnné crystal (M i , F, V ) t by [13, 5.11-5.12].
Set M ( C,Z) to be the evaluation of M on the trivial thickening of ( C, Z) and set Ω to be the kernel of F on M ( C,Z) ; then Ω is a vector bundle on C, called the Hodge vector bundle.(Kato-Trihan obtain the dual version of this in [9, 5.1], especially Lemma 5.3 of loc.cit..) 12 Similarly, we can construct the Hodge bundle Ω i of each (M i , F ), which will be a line bundle on C.Moreover, there is a short exact sequence: We have an isomorphism of vector bundles on C: As each G C,i has non-empty supersingular locus, it follows that the Hasse invariant associated to G C,i : Hasse is nonzero, which implies that Ω i is a positive degree line bundle on C. As G t C,i is supersingular exactly when G C,i is supersingular, we deduce that Ψ i is also positive.Therefore, Ω = Ω B/ C splits as the direct sum of 8h positive line bundles.In particular, we have shown (i).We further note that Ω is isomorphic to the Hodge line bundle associated to the Néron Model of B C → C by [14, A.11] (this was first proven in [9]).
For the next step: Faltings-Chai have constructed the following Kodaira-Spencer map [7, Ch.III, Corollary 9.8]: extending the usual Kodaira-Spencer map over C. As B admits a principal polarization, we have that B C ∼ = B t C , and hence B C ∼ = Bt C , as both are simply the respective Néron models.Therefore, we may equivalently write Equation 2.2 as: ).Under the decomposition: and after restricting to C, the above θ| C is just the sum of the Kodaira-Spencer maps for each G i and G t i : These were constructed to be nonzero, as both G i and G t i are generically versally deformed; therefore, the Kodaira-Spencer map of sheaves is an injective map of coherent sheaves.Therefore (ii) is shown.
Fortunately, (iii) is an easy corollary of (ii).Indeed, taking degrees, we see that: ), from which the inequality follows immediately.
Finally, let us prove (iv).By moduli theory, we have a map C → A 8h,1,N , where the latter is a fine moduli scheme.As C is a smooth curve and the Baily-Borel compactification A * 8h,1,N is proper, it follows that we get an extension: Finally, the argument that the Hodge line bundle on A * 8h,1,N pulls back under λ to det(Ω B/ C ) is given in the text surrounding [14,Equations 3.4,3.5].(While the argument in loc.cit. is only written for N = ℓ, it generalizes verbatim to the matter at hand.Indeed, the argument is an easy corollary of [7, Ch.V, Theorem 2.5].)

The moduli spaces
We work in the following situation.Let K be number field, let N ≥ 1, set S := Spec(O K [1/N]), and let X/S be a smooth projective curve, let D ⊂ X be a relative reduced divisor, and let U denote the open complement.
Let ℓ be a prime number and g ≥ 1 an integer.We again denote by A * g,1,ℓ 3 the Baily-Borel compactification of A g,1,ℓ 3 , which is defined over Spec(Z[1/ℓ]).This moduli space has a natural ample line bundle, the Hodge line bundle, which we denote by α.Definition 3.1.Fix a positive integer b.Denote by H the following contravariant pseudofunctor from the category of S schemes to the 2-category of groupoids.The value H(T ) for an S-scheme T is the groupoid of triples (Y, ϕ, λ), that fit into a diagram as follows: • ϕ is finite morphism, of degree at most ≤ |GL 2g (Z/ℓ 3 Z)|, and étale over U; • there exists some cusp ∞ of Y(T ) (lying over a point of D(T ) of X(T )) that is sent to a 0-dimensional cusp of A * g,1,ℓ 3 × T ; • and the degree of the pulled-back Hodge line bundle λ * (α) on every geometric fiber of Y is ≤ b, with the natural notion of isomorphism: if (Y, ϕ, λ) and (Y ′ , ϕ ′ , λ ′ ) are elements of H(T ), then an isomorphism between them is an X T -isomorphism Y → Y ′ that intertwines ϕ and ϕ ′ as well as λ and λ ′ .
Colloquially, the functor H parameterizes finite covers Y of X equipped with a (principally polarized) abelian scheme of dimension g with trivial ℓ 3 level structure, such that the induced map Y → A * g,1,ℓ 3 has "degree" bounded by b.
, which corresponds to the condition that sends λ(ϕ −1 (U)) ⊂ A g,1,ℓ 3 , i.e., that ϕ −1 (U) is sent inside of the moduli space of abelian varieties.It follows that H is finite type Deligne-Mumford stack over S. By further imposing the condition that the map λ sends at least one point in the boundary divisor to a zero dimensional cusp of the Baily-Borel compactification, H is a closed substack of H , which is again finite type.Now, let L be a lisse Q ℓ -sheaf as in Theorem 1.4.There exists an ℓ-adic local field M/Q ℓ such that the associated representation factors through the ring of integers O M : Abusing notation, we call the induced lisse O M -sheaf L. Denote by π M the uniformizer of M and κ M the residue field of M. Definition 3.3.Fix i ≥ 1 and a lattice L as above.Let Hi denote the following contravariant pseudofunctor from S-schemes to groupoids: the value Hi (T ) on an S-scheme T is the collection of quadruples (Y, ϕ, λ, ψ), where (Y, ϕ, λ) ∈ H(T ), and ψ is the following extra piece of data.As λ : W := ϕ −1 (U) → A g,1,ℓ 3 , there is a principally polarized abelian scheme f : A W → W (with trivial ℓ 3 -torsion).Then is a map of étale torsion sheaves on W whose reduction modulo π M -reduction is nonzero.In other words, im(ψ There is an obvious notion of isomorphism of two such quadruples.
The pseudo-functor Hi is actually a stack in the étale topology.This follows from the following two properties.Let T be a scheme: • There exists an internal Hom in the category of torsion locally constant abelian étale sheaves on T .• If ψ : F → G is a map of torsion, locally constant étale sheaves of O M modules on T , then the property that ψ(F ) ⊂ π M (G) may be checked on an étale cover.There are natural transformations of pseudo-functors Hj → Hi for any j > i.We claim that Hi represents a finite type Deligne-Mumford stack over S. To prove this, it suffices to prove that the natural transformation of pseudo-functors Hi → H is representable by a scheme.
Let T be an S-scheme, and t := (Y, ϕ, λ) ∈ H(T ).Then we have the following pullback square: Then T × H Hi has the following description.There are two natural ℓ i -torsion étale sheaves on Y: ϕ * (L/ℓ i ) (which has O M /ℓ i -rank 2), and R 1 f * O M /ℓ i (which has O M /ℓ irank 2g).Then T × H Hi corresponds to the (finite) set of injective maps of sheaves of abelian groups: ψ : ϕ * (L/ℓ i ) ֒→ R 1 f * O M /ℓ i .This finite set is canonically a scheme.It follows that Hi → H is relatively representable, and hence Hi is represented by a Deligne-Mumford stack of finite type over S. Proposition 3.4.The natural map "forget ψ": Hi → H is finite.
Proof.It is obviously quasi-finite because, as argued above, if we fix i, then there are only finitely many choices for ψ.To prove it is finite, we show that it is proper.
As both Hi and H are of finite type over S = Spec(O K [1/N]), it suffices to simply check the valuative criterion for properness.Let R be a discrete valuation ring with fraction field F .Suppose we have (Y, ϕ, λ) ∈ H(R) and (Y F , ϕ F , λ F , ψ F ) ∈ Hi (F ).Therefore we have a principally polarized abelian scheme f ′ : A W → W (of dimension g, with trivial ℓ 3 torsion), together with a map of torsion étale sheaves over W F whose reduction modulo π M is nontrivial: Note the following.If one has two finite étale sheaves on an irreducible normal scheme, and a morphism between them over the generic point, then that morphism uniquely extends to the whole scheme.(Here, we are closely following [25, Proof of Lemma 23].)Therefore, ψ F extends to a ψ on all of U ′ R , and we have verified the valuative criterion for properness.Definition 3.5.For i ≥ 1, set As Hi → H is relatively representable and finite (Proposition 3.4), it is universally closed.Therefore H i is a closed subset of |H|, which we may equip with the induced reduced substack structure [26,Tag 0508].According the natural transformations Hj → Hi , the sequence of closed subsets are descending.Set: which is also equipped with the reduced induced substack structure.Then H i and H ∞ are Deligne-Mumford stacks of finite type over S = Spec(O K [1/N]) for all i ≥ 1.
Lemma 3.6.Let T be an S-scheme and let (Y, ϕ, λ) ∈ H(T ).Then the following conditions are equivalent: (1) (Y, ϕ, λ) ∈ H ∞ (T ); (2) there exists an injection we get the desire injection.(Note that both are lisse O M -sheaves, so tensoring with M yields an injective map.) , there exists some integer ι such that Then the map π ι M τ is an injection from ϕ * (L) to R 1 f * O M .This is equivalent to saying that the reduction modulo π M of ψ is nontrivial.Denote ψ i = ψ mod (π i M ) for each i > 0, Since ψ i mod (π M ) = ψ mod (π M ) = 0, the quadruple (Y, ϕ, λ, ψ i ) ∈ Hi .Thus (Y, ϕ, λ) ∈ ∩ ∞ i=1 H i = H ∞ .(1) ⇒ (3): (This is the main content of the Lemma.)Since (Y, ϕ, λ) ∈ H ∞ (T ), for each i > 0, there exists a map which is nontrivial modulo π M .In general, the ψ ′ i 's do not form a compatible sequence, i.e., it is possible that there exists j > i with the following property: ψ ′ j mod (π i M ) ≡ ψ ′ i .Therefore, one can not directly take projective limits to find our desired map ϕ * (L) → R 1 f * O M .However, we claim we may derive a compatible sequence from ψ ′ i as follows.
Consider the subset in the finite set M reductions ψ ′ i mod (π r+1 M ): Again by the pigeonhole principle, there exists a nontrivial map ψ r+1 ∈ Σ r+1 and an infinite subset N r+1 ⊂ N r such that ψ ′ i mod (π r+1 M ) = ψ r+1 for any i ∈ N r+1 .Iteratively, we find a sequence ψ 1 , ψ 2 , • • • satisfying ψ j mod (π i M ) = ψ i for each j > i. Taking projective limits and tensoring with M, one gets a nonzero map Since L ⊗ M is irreducible, ψ is injective.

Rigidity
In this section, we prove the following.Recall that S = Spec(O K [1/N]) Lemma 4.1.Let H/S be as in Section 3.Then, after potentially increasing N (equivalently, replacing S by a non-empty Zariski open subset), the relative dimension of H/S is 0.
Proof.We have shown that H/S is a finite type Deligne-Mumford stack.To show the desired result, it suffices to show that that if K ֒→ C is an embedding, then H C has dimension 0. Equivalently, we want to show that if A U C → U C is a principally polarized abelian scheme that is totally degenerate at at least one cusp, then it is rigid.This immediately follows from Theorem 8.6 together with Lemma 3.4 and the following text of [23].

The proof
Proof of Theorem 1.4.First, assume that L has bad, unipotent reduction around every cusp.Let T 1 be the set of those prime p of O K with the following properties: the underlying prime p splits completely in E, and p > max(N, ℓ 3 ).This is an infinite set by the Cebotarev density theorem.Let L p be the restriction of L to U p .Then L p is irreducible by exactly the same argument as that of the first paragraph of [25, Proof of Lemma 24, p. 2053].
There are only finitely may subfields of E. It follows from the pigeonhole principle that there exists a subfield F ⊂ E such that there exists infinitely many primes p ∈ T 1 such that L p has Frobenius traces in F ⊂ E. Call the collection of such primes T 2 ⊂ T 1 .Let

Definition 2 . 3 .
Maintain notation as in Setup 2.1.Let G C be a p-divisible group on C. We say G C has strong strict semistable reduction along Z if • G C has semistable reduction along Z [27, Definition 4.2] (which is based on semistable reduction in the sense of de Jong [2, Definition 2.2]); and

Proposition 2 . 4 .
Maintain notation as in Setup 2.1.Let G C be a height 2, dimension 1 p-divisible group on C with strong strict semistable reduction along Z. Suppose further that D Fq((z∞))) .(Here, the meaning of G µ Spec(Fq((z∞))) and G f Spec(Fq((z∞))) is given as in [2, Definition 2.2].)However, by definition of semi-stability, H C | Spec(Fq((z∞))) therefore extends to a pdivisible group over Spec(F q [[z ∞ ]]).Ranging over all points ∞ ∈ Z, we see thatD(H C ) ⊗ Q p ∈ F-Isoc(C) in fact extends to an F -isocrystal on F-Isoc( C): therefore D(H C )⊗Q p ∈ F-Isoc † (C).However, this yields a sub-object (in F-Isoc † (C)) of D(G C ) ⊗ Q p , contra-dicting the absolute irreducibility of the hypothesis.Therefore G C has a non-empty supersingular locus.Now, suppose that G C → C is not generically versally deformed, i.e., that KS = 0 identically on C. Then by[28, Theorem 6.1], there is a p-divisible group (G 1 ) C → C such that (G 1 ) (p) C ∼ = G C , i.e., the Frobenius twist of (G 1 ) C is isomorphic to G C .The p-divisible groups G C and (G 1 ) C are isogenous.If the Kodaira-Spencer map for (G 1 ) C is nonzero, we may stop.Otherwise, we may apply[28, Theorem 6.1]  again to find a p-divisible group (X,D)/S parametrizing finite tame covers Y → X of degree ≤ c such that:• Y/S has geometrically connected fibers and• Y → X is étale over U := X \ Dis represented by a finite-type Deligne-Mumford stack over S.There is a natural map Cov ≤c (X,D)/S → M := k bounded M k , which is the map that sends a cover Y → X to the underlying curve Y. Here, the notation M k stands for the moduli space of genus k curves.Denote by C k → M k the universal curve and by C := C k , which has a natural map C → M .Then consider H , the open substack of the 2-fiber product:

Σ 1 =
Hom(ϕ * (L)/π M , R 1 f * O M /π M ) consisting of all modulo π M reductions of ψ ′ i : {ψ ′ i mod (π M ) | i ≥ 1}.By the pigeonhole principle, there exists a nontrivial map ψ 1 ∈ Σ 1 and an infinite subsetN 1 ⊂ N such that ψ ′ i mod (π M ) = ψ 1 for any i ∈ N 1 .Suppose we have constructed a compatible sequence ψ 1 , ψ 2 , • • • , ψ r and an infinite subset N r ⊂ N satisfying ψ ′ i mod (π j M ) = ψ j ∈ Hom(ϕ * (L)/π j M , R 1 f * O M /π j M ) for any i ∈ N r and j ∈ {1, 2, • • • , r}.Then we consider the subset in the finite setΣ r+1 = {ρ ∈ Hom(ϕ * (L)/π r+1 M , R 1 f * O M /π r+1 M ) | ρ mod π r M = ψ r } consisting of all modulo π r+1 generically ordinary with a non-empty supersingular locus; and (2) there exists an isogenous p-divisible group H C → G C that is generically versally deformed (in the sense of [12, Defintions 8.1, 8.2]). 8 1,N , where the latter denotes the Baily-Borel compactification.Then the Hodge line bundle α on A * 8h,1,N pulls back to det(Ω B/ C ). Proof.Construct A C → C as in Corollary 2.5.Again, by Grothendieck's criterion for semistable reduction of abelian varieties, A C → C must have semistable reduction.Set B C := (A C × A t C ) 4 ; then by a result of Zarhin [19, Chapitre IX, Lemme 1.1, p. 205], B C → C is principally polarized.Moreover, it clearly has semistable reduction.From the construction, and the fact that Proposition 3.2.After potentially increasing N (equivalently, replacing S by a nonempty open subscheme), the functor H is represented by a finite type Deligne-Mumford stack over S.Proof.By increasing N, we can assume that all covers ϕ : Y → X that occur in the definition of H are tamely ramified at the cusps.More precisely, this will hold for anyN > |GL 2g (Z/ℓ 3 Z)|.It follows from the theory of the Hilbert scheme that for any noetherian scheme T and for any relative smooth proper curve Y/T with geometrically connected fibers, the functor Hom ≤b T (Y, A * g,1,ℓ 3 ) parametrizing maps λ such that the deg(λ * (α)) ≤ b for every geometric fiber is represented by a finite-type T -scheme.(In particular, this holds true even if T is not connected and the genus of the fibers varies on different connected components.)It follows that if T is a Deligne-Mumford stack, of finite type over S, and Y/T is a smooth, proper curve with geometrically connected fibers, then the same functor Hom ≤b T (Y, A * g,1,ℓ 3 ) is represented by a finite type Deligne-Mumford stack over T .On the other hand, the theory of the Hurwitz scheme implies that the functor Cov c