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This result is obtained as a corollary of the following geometric statement for the p-primary torsion of abelian varieties over function fields of curves


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Abstract. Letkbe a field finitely generated overQand denote by Γk its absolute Galois group. LetAbe an abelian variety overk. Given a primepand a characterχ: ΓkZp, define thetwistedp-primary torsion associated withχto be the Γk-module:

A[p](χ) :={T A[p]|σT =χ(σ)T , σΓk},

whereA[p] denotes thep-primary torsion ofA(k). Say thatχisnon-Tateif it does not appear as a subrep- resentation of thep-adic representation associated with an abelian variety overk. (This holds, for instance, if χis the trivial or thep-adic cyclotomic character.) Ifχis non-Tate,A[p](χ) is always finite. Our main result is about the uniform boundedness ofA[p](χ) whenAvaries in a 1-dimensional family. More precisely, ifSis a smooth, geometrically connected curve overkandAis an abelian scheme overS, then there exists an integer N:=N(A, S, k, p, χ), such thatAs[p](χ)As[pN] holds for anysS(k).

This result is obtained as a corollary of the following geometric statement for the p-primary torsion of abelian varieties over function fields of curves. LetKbe the function field of a smooth, geometrically connected curve over an algebraically closed field of characteristic 0 and let Abe an abelian variety over K. Assume for simplicity thatAcontains no nontrivial isotrivial subvariety. Then, for anyc0, there exists an integer N:=N(c, A, S, k, p)0 such thatA[p](K0)A[pN] for all finite extensionK0/KwithK0of genusc.

Our result about the uniform boundedness of twistedp-primary torsion for thep-adic cyclotomic character together with certain descent methods yields a proof of the 1-dimensional case of the modular tower conjecture, which was, actually, the original motivation for this work. The modular tower conjecture is a conjecture from regular inverse Galois theory posed by M. Fried in the early 1990s, (a generalized variant of) which roughly states that if (Hn+1Hn) is a projective system of moduli spaces classifying G-covers of genusgcurves with groupGn withr ramification points, then, for any fieldkfinitely generated over Q,Hn(k) = fornlarge enough, provided the projective limit of (Gn+1Gn) admits an open subgroup which is free pro-p of finite positive rank.

2000Mathematics Subject Classification.Primary: 11G10, 11G30, 14H30; Secondary: 14D22, 14G32.

1. introduction

Fix a primep and letkbe a field of characteristicq 6=p and letS →k be a smooth, geometrically connected curve over k of type (g, r) (that is such that there exists a proper, smooth, geometrically connected, genus g curve S → k over k and a degree r etale divisor D ⊂ S such that S = SrD).

Denote by η:k(S)→S the generic point ofS.

Given a geometric generic point s : Ω → S denote by Π := π1(S, s) the etale fundamental group of S. Recall that, in our situation, it is simply the Galois group of the maximal algebraic extension of k(S) in Ω which is unramified on S; in particular, it is a quotient of the absolute Galois group Γη := Gal(k(S)sep/k(S)), wherek(S)sep stands for the separable closure of k(S) in Ω.

Now, fix an abelian schemeA →S with generic fiber Aη →η of dimensiond. We refer to section 2.1 for the notation used for abelian varieties. Since A → S is an abelian scheme and q 6= p, the natural action of Γη on the Tate module Tp(Aη) factors through Π. This, in turn, defines an action of Π overAη[pn],n≥0 hence overAη[p]. Finally, for anyv∈Aη[p], write Πv for the stabilizer of v under Π; it is a closed subgroup of Π of index ≤ |hvi|2d, hence it is also open in Π, and, by Galois theory, corresponds to a connected finite etale cover Sv → S (defined over a finite extension kv/k).

Similarly, write Πhvi for the stabilizer of hvi under Π and Shvi → S for the resulting connected finite etale cover (defined over a finite extension khvi/k). The inclusion of open subgroups Πv ⊂Πhvi ⊂ Π



yields a commutative diagram of finite etale covers:

Sv //




We will write gv and ghvi for the genus of the smooth compactifications of Sv×kvk and Shvi×khvik, respectively.

Eventually, denote by (Aη)0 the largest abelian subvariety ofAη which is isogenous to an isotrivial abelian variety (see section 2).

We can now state the main geometric result of this paper:

Theorem 1.1. Assume that k is algebraically closed. Then, for any c ≥ 0, there exists an integer N := N(c, A, S, k, p) ≥ 0 such that for all v ∈ Aη[p] either gv ≥ c or pNv ∈ (Aη)0. The same statement holds if gv is replaced by ghvi.

Say that a character χ : Γk → Zp is non-Tate if it does not appear as a subrepresentation of the p-adic representation associated with an abelian variety over k. This holds, for instance, if χ is the trivial or the p-adic cyclotomic character, providedk is finitely generated over the prime field. When χis non-Tate,A[p](χ) is always finite. Theorem 1.1 yields the following corollary about the uniform boundedness of A[p](χ) when A varies in a 1-dimensional family.

Corollary 1.2. Assume that k is finitely generated over Q and let χ : Γk → Zp be a non-Tate character. Then there exists an integer N := N(A, S, k, p, χ), such that, for any s ∈ S(k), the Γk- module As[p](χ) :={T ∈As[p]|σT =χ(σ)T, σ∈Γk} is contained inAs[pN].

The strategy of the proof of theorem 1.1 is as follows. First, we show that it is enough to prove the following statement (Theorem 4.1): Assume that Aη contains no nontrivial abelian subvariety isoge- nous to an isotrivial abelian variety, and, for any v∈Tp(Aη), set vn:=v modpnTp(Aη)∈Aη[pn], n≥0. Then gvn → ∞ and ghvni → ∞.

Then, to any v ∈ Tp(Aη) associate the Π-module M := Qp[Π]v∩Tp(Aη) and denote by Π(n) the kernel of the reduction morphism GL(M) GL(M/pn). The inclusion of open subgroups Π(n) ⊂Πvn ⊂ Π corresponds to a sequence of finite etale covers S(n) → Svn → S, with S(n) → S Galois, which makes it easier to handle. We first prove that the genusg(n) ofS(n) goes to infinity with n. This is achieved by using the fact that the image of Π in GL(M) cannot be almost-abelian (section 2.3). Then, using the Riemann-Hurwitz formula, we compare g(n) and gvn and show that the “in- creasing rate” ofgvn is “close enough” to that ofg(n) forn0. This relies on the asymptotic bound given by J. Oesterl´e for the number of points on reduction modulopn of p-adic analytic subspaces of Zmp . The proof of the variant forghvi is along the same lines but working with the projectivization of the representation of Π over M.

To deduce corollary 1.2 from theorem 1.1, we introduce a sequence of (disconnected) finite etale coversSn+1,χ→ Sn,χofS with the property thatk-rational pointssn:k→ Sn,χlying aboves:k→S correspond to elements of order exactly pn inAs[p](χ). It is thus enough to prove that Sn,χ(k) =∅ for n 0. According to theorem 1.1, Sn,χ can be written as a disjoint union Sn,χ = Sn,χ(1)`

Sn,χ(2), where Sn,χ(1) corresponds to the “isotrivial part” of Aη[pn] and Sn,χ(2) to the “non-isotrivial” one. On one hand, we show that Sn,χ(1)(k) =∅ forn0, and, on the other hand, from theorem 1.1, the genus of each component of Sn,χ(2) is larger than 2 for n 0. Applying the Mordell conjecture (Faltings’

finiteness theorem) and the definition of non-Tatep-adic character then yields the result.

Our original motivation for corollary 1.2 was to prove the 1-dimensional case of the modular tower conjecture. The modular tower conjecture is a conjecture from regular inverse Galois theory posed by M. Fried in the early 1990s (cf. [Fr95]), (a generalized variant of) which roughly states that if


(Hn+1→Hn) is a projective system of moduli spaces classifying G-covers of genusgcurves with group Gnand ramification indicesen,1, . . . , en,r, then, for any fieldkfinitely generated overQ,Hn(k) =∅for nlarge enough, provided the projective limit of (Gn+1→Gn) admits an open subgroup which is free pro-p of finite positive rank. As theHnhave dimension 3g−3 +r (under the hyperbolicity condition 2−2g−r <0), the 1-dimensional case corresponds to (g, r) = (0,4), (1,1). (The 0-dimensional case corresponds to (g, r) = (0,3) and is easy to treat. See lemma 5.11.) In that case, we thus have (theorem 5.12): The modular tower conjecture holds for(g, r) = (0,4), (1,1). A typical example of the modular tower conjecture is when one takes for Gn the dihedral groupD2pn of order 2pn, (g, r) = (0,2s), and en,1 = · · · = en,2s = 2. Then, roughly speaking, Hn classifies jacobians of genus s−1 hyperelliptic curve with a torsion point of order excatly pn. In particular, for s = 2, (Hn+1 → Hn) is just the projective system of modular curves (Y1(pn+1)→Y1(pn)) and, in that special case, the modular tower conjecture can be deduced from the Mordell conjecture (Faltings’ finiteness theorem) and the fact that lim

←− Y1(pn)(k) =∅. This is the basic idea (essentially due to Manin [Ma69]) hidden behind our proof of corollary 1.2 and its application to the modular tower conjecture. In the general case, one has to distinguish between the fine and coarse moduli situations. If Hn is a fine moduli scheme for n≥0, then one can apply directly corollary 1.2 to the p-adic cyclotomic character and the jacobian Pic0C

0|H0 →H0 of the universal curve C0 →H0. WhenHn is a mere coarse moduli scheme for n≥0, the idea is, roughly, to use descent techniques to construct an auxilliary tower ( ˜Hn+1 → H˜n) of fine moduli schemes with the property that ˜Hn(k)6=∅ as soon asHn(k)6=∅.

The paper is organized as follows. In section 2, we list the notation used for abelian varieties (section 2.1) and collect technical preliminaries about non-Tate characters (section 2.2) and almost- abelian p-adic representations (section 2.3). In section 3, we prove a result (theorem 3.1) concerning reduction modulopn of certainp-adic analytic spaces, which is the technical core of our proofs of the main results. In section 4, the proofs of theorem 1.1 corollary 1.2 are carried out in sections 4.1 and 4.2, respectively. Eventually, in section 5, we convey the proof of the 1-dimensonal modular tower conjecture. After recalling the basic notion about G-curves and their stacks (section 5.1), we describe carefully our descent techniques (section 5.2) and apply them together with corollary 1.2 (section 5.3).

Remark 1.3. In [Fr06], Fried outlined a proof, which is different from ours, of (the original version of) the modular tower conjecture for (g, r) = (0,4).

2. Preliminaries on abelian schemes

2.1. Notation for abelian varieties. Fix a prime p and let k be a field of characteristic q 6= p.

Given an abelian varietyA→k, we will use the following (classical) notation:

A[pn] :=ker([pn] :A(k)→A(k)) for the kernel of the multiplication-by-pn endomorphism;

A[p] := [


A[pn] for thep-primary torsion ofA(k).

Tp(A) := lim

←− A[pn] for thep-adic Tate module ofA;


Tp(A) :=Tp(A)rpTp(A) = lim

←− A[pn]; Vp(A) :=Tp(A)⊗ZpQp;

Now, if A → η is defined over the function field η of a k-curve S → k, we will write A0 for the largest abelian subvariety ofAwhich is isogeneous to an isotrivial abelian variety. (Herem we say that an abelian variety B →η is isotrivial, ifB×ηη descends to an abelian variety over k.) Equivalently, A0 is the largest abelian subvariety of A which is a homomorphic image of an isotrivial abelian variety. Indeed, if A1 → A2 is an epimorphism of abelian varieties, then, by Poincar´e’s complete reducibility theorem,A2 is isogeneous to an abelian subvariety ofA1. But as any abelian subvariety of an isotrivial abelian variety is again isotrivial, ifA1 is isotrivial then A2 is isogeneous to an isotrivial abelian subvariety ofA1. In particular, ifAi⊂Ais isogeneous to an isotrivial abelian variety,i= 1,2, then so is A1+A2⊂A (as a homomorphic image ofA1×A2); thus A0 is well-defined. (When khas


characteristic 0, any abelian variety isogeneous to an isotrivial abelian variety is again isotrivial, so A0 is simply the largest isotrivial subvariety of A.) We define A0 to be the quotient abelian variety A/A0. Then (again by Poincar´e’s complete reducibility theorem) we have (A0)0= 0.

2.2. Non-Tate characters. Fix a prime p and let k be a field of characteristic q 6= p. Let χ : Γk →Zp be a character. Say that χ is non-Tate if it does not appear as a subrepresentation of the p-adic representation associated with an abelian variety over k. When k is finitely generated over its prime field, the trivial character and thep-adic cyclotomic character are typical examples of non-Tate characters.

Lemma 2.1. For any finitely generated extension k0 of k, χ : Γk → Zp is non-Tate if and only if χ|Γ

k0 : Γk0 →Zp is non-Tate.

Proof. First, the ‘if’ part is trivial. Indeed, ifχ appears in Tp(A) for some abelian variety A overk, χ|Γ

k0 appears in Tp(A×kk0). Next, to show the ‘only if’ part, suppose that χ|Γ

k0 appears in Tp(A0) for some abelian varietyA0 overk0.

Assume first that k0 is a finite extension of k. In this case, one can easily reduce the problem to the following two cases: k0/k is separable; and k0/k is purely inseparable (for q > 0). In the former case, defineAto be the Weil restrictionResk0/k(A0), which is an abelian variety overkwith dimension [k0:k]dim(A0). Then the adjoint property of Weil restriction implies thatχappears inTp(A). Indeed, more specifically, let Mn denote the finite etale commutative group scheme over k that corresponds to the Galois moduleZ/pnn) (i.e., the moduleZ/pn on which Γk acts via χn). Then

HomΓk(Zp(χ), Tp(A)) = lim

←−Homk(Mn, A) = lim

←−Homk0(Mn×kk0, A0) = HomΓk0(Zp(χ|Γ

k0), Tp(A0)).

In the latter case, take N ≥0 such that (k0)qN ⊂k and set A:= A0×k0k, where k0 → k is given by a7→ aqN. Then, by using the fact that the qNth power map k → k induces the identity on Γk, one can show thatχ appears inTp(A).

Finally, assume that k0 is an arbitrary finitely generated extension of k and that χ|Γ

k0 appears in Tp(A0) for some abelian variety A0 over k0. Then, taking a model of A0 → k0 over the spectrum of a finitely generated k-subalgebra of k0 whose fraction field coincides with k0 and specializing it at a closed point, one can show that there exist a finite extension k00 of k and an abelian variety A00 over k00 such thatχ|Γ

k00 appears inTp(A00). Thus, by the preceding argument,χappears inTp(A) for some abelian variety overk.

2.3. Almost-abelian p-adic representations. Fix a prime p. Let V be a finite-dimensional Qp- vector space. For a property P of subgroups of AutQp(V) (e.g., trivial, abelian, unipotent, pro-p, etc.), say that a subgroupGof AutQp(V) isalmost-P if there exists a closed subgroupH ⊂Gof finite index such thatH hasP. Further, given a group Π and a representationρ: Π→AutQp(V), say that ρ : Π → AutQp(V) is almost-P if the image of ρ is almost-P (though the standard terminology for

‘almost-unipotent’ may be ‘quasi-unipotent’).

Let kbe an algebraically closed field of characteristic q 6=p and let S →k be a connected normal scheme of finite type overk. Denote by η:k(S)→S the generic point ofS. LetA→S be an abelian scheme with generic fiberAη →η, and consider the natural p-adic representation on the Tate module ρ:π1(S)→AutZp(Tp(Aη))⊂AutQp(Vp(Aη)).

Proposition 2.2. Let M be a π1(S)-submodule of Tp(Aη) and and consider the corresponding repre- sentation ρM1(S)→AutZp(M)⊂AutQp(M⊗ZpQp). Then the following are all equivalent.

(1) ρM is almost-trivial (i.e., ρM has finite image).

(2) ρM is almost-abelian.

(3) ρM is almost-unipotent.

(4) M ⊂Tp((Aη)0).

Proof. (1)⇒(2). Clear.


(2)⇒(3). First observe AutZp(M)'GLm(Zp), wheremis the rank of the freeZp-moduleM. Thus, from the short exact sequence

1→1 +pMm(Zp)→GLm(Zp)→GLm(Z/p)→1,

the group AutZp(M) is almost-pro-p. This, together with the assumption that ρM is almost-abelian, assures that one may assume that ρM1(S)) is an abelian pro-p group, up to replacing S by a finite etale cover.

SinceS is of finite type overk, there exists a subfield k1 ofk, finitely generated over its prime field, such that A → S → k admits a model A1 → S1 → k1, whereS1 → k1 is a geometrically connected, geometrically normal scheme of finite type over k1 and A1 → S1 is an abelian scheme. Denote by η1 : k1(S1) → S1 the generic point of S1 and by A1,η1 → η1 the generic fiber of A1 → S1. One has a canonical surjection π1(S) π1((S1)k

1) ([SGA1, Exp. X, proof of Cor. 1.6]). At the level of Tate modules, one has a canonical isomorphism Tp(Aη) ˜→Tp(A1,η1) which can be taken to be compatible with the actions of π1(S) and π1((S1)k

1). Thus, one may assume that k = k1. Here, observe that Tp(Aη) =Tp(A1,η1) admits a natural action ofπ1(S1), extending the action ofπ1(S) =π1((S1)k).

Denote by ∆ the maximal abelian pro-p quotient of π1(S) and by N the kernel of the natural surjectionπ1(S)∆. Set Π :=π1(S1)/N. Thus, one gets the following exact sequence

1→∆→Π→Γk1 →1.

Since ∆ is abelian, this exact sequence induces a natural action of Γk1 on ∆.

Lemma 2.3. For any finite extensionk01 of k1, the coinvariant quotient ∆Γk0 1

is finite.

Proof. This is essentially widely known, and follows, for example, from the fact that the (Frobenius) weights that may appear inHet1(S,Qp) are 1 and 2.

As ρM1(S)) is an abelian pro-p group, ρM : π1(S) → AutZp(M) factors through ∆, or, equiv- alently, M is contained in Tp(Aη)N. Now, up to replacing M by Tp(Aη)N, one may assume that M ⊂ Tp(Aη) is a π1(S1)-submodule and that the resulting representation ρM1(S1) → AutZp(M) factors through Π.

Now, define ∆ and Π to be the images of ∆ and Π in AutZp(M). SetR0:=Zp[∆]⊂EndZp(M) and R := R0Zp Qp. As R0 is a free Zp-module, the canonical morphism R0 → R is injective, hence so is Rred0 →Rred, where Rred0 =R0/√

0R0 and Rred :=R/√

0R. We shall denote the natural surjection R Rred by f 7→ fred. The action of Π by conjugation on ∆ extends to one onR0 by Zp-linearity and to one on R by Qp-linearity. Note that Π acts onR not only as Qp-module automorphisms but also as Qp-algebra automorphisms. In particular, the action of Π on R induces one on Rred. Denote by i: Π Π → AutQp-alg(R) and ired: Π →AutQp-alg(Rred) the resulting representations. Observe thati (hence alsoired) factors through ΠΓk1, since ∆ is abelian.

Since Rred is a finite reduced commutativeQp-algebra, it is isomorphic to a product Q

1≤i≤rLi of finite field extensionsLi/Qp,i= 1, ..., rand itsQp-algebra automorphism group is finite. In particular, iredhas finite image. So, up to replacingk1 by a finite extension, one may assume thatired is trivial.

Then, if we denote the image of ∆ in (Rred0 ) ⊂(Rred) by ∆red, it follows that the natural surjection

∆∆redfactors through ∆Γk

1. Thus, by lemma 2.3, ∆redis finite.

Now, up to replacing S by a finite etale cover, ∆ has trivial image in (Rred), or, equivalently, has unipotent image in R⊂Aut(M⊗ZpQp), as desired.

(3)⇒(4). Up to replacingS by a finite etale cover, we may assume thatρM is unipotent. Denote by M0 the image ofM inTp(Aη)/Tp((Aη)0) =Tp((Aη)0), and suppose M0 6={0}. Then ρM01(S) → Aut(M0) is also unipotent. From this, there exists a v ∈M0r{0} such that π1(S) acts trivially on v. In particular, this forces thep-primary torsion of (Aη)0(η) to be infinite. But this contradicts the fact that (Aη)0(η) is a finitely generated abelian group by the Lang-N´eron theorem ([LN59]). Thus, M0={0}, as desired.

(4)⇒(1). Immediate from the definition of (Aη)0. Corollary 2.4. The following are all equivalent.


(1) Tp(Aη) admits no nonzeroπ1(S)-submoduleM withρM almost-trivial.

(2) Tp(Aη) admits no nonzeroπ1(S)-submoduleM withρM almost-abelian.

(3) Tp(Aη) admits no nonzeroπ1(S)-submoduleM withρM almost-unipotent.

(4) (Aη)0 = 0.

Proof. (4) is equivalent to Tp((Aη)0) = {0}, hence to saying that Tp(Aη) admits no nonzero π1(S)- submoduleM withM ⊂Tp((Aη)0). Thus, corollary 2.4 immediately follows from proposition 2.2.

Corollary 2.5. Consider the following conditions:

(1) ρ is almost-trivial.

(2) ρ is almost-abelian.

(3) ρ is almost-unipotent.

(4) (Aη)0 =Aη. (5) Aη is isotrivial.

Then we have (1) ⇔ (2) ⇔ (3) ⇔ (4) ⇐ (5). (In particular, if π1(S) itself is almost-abelian (i.e., admits an abelian open subgroup), then (4) automatically holds.)

If, morever,q = 0, (1)–(5) are all equivalent.

Proof. Applying proposition 2.2 toM =Tp(Aη), one gets the equivalence of (1)–(4). For (5)⇒(4) and (4)⇒(5) for q= 0, see 2.1.

Remark 2.6. In corollary 2.5, the equivalences (1)⇔(4) and (1)⇔(5) for q = 0 directly follow from [G66, Prop. 4.3 and Prop. 4.4].

Remark 2.7. LetX→S be a proper, smooth, geometrically connected curve and apply corollary 2.5 toA= Pic0X/S, the jacobian ofX→S. Further, consider the following condition: (6) Xη is isotrivial.

Then we have (5)⇔(6). Indeed, (6)⇒(5) is clear and (5)⇒(6) essentially follows from a version of Torelli’s theorem. We omit the details.

3. Reduction modulo pn of p-adic analytic homogeneous spaces

The aim of this section is to prove theorem 3.1 concerning reduction modulo pn of certain p-adic analytic homogeneous spaces, which is the technical core of our proofs of the main results in§4. The readers who would like to start the proofs of the main results quickly may skip this section, after glancing at the statement of theorem 3.1.

Let p be a prime. Let M be a free Zp-module of rank m ≥ 1 and set Mn := M/pn, n ≥ 0, and W := M ⊗Zp Qp. Set GL(M) := AutZp(M) and GL(Mn) := AutZ/pn(Mn), n ≥ 0. Let M Mn, v 7→vn denote reduction modulo pnM and GL(M)GL(Mn), g 7→gn the induced morphism.

As projective versions of these, set P(M) := (M rpM)/Zp and P(Mn) := (MnrpMn)/(Z/pn), n ≥ 0. Set PGL(M) := GL(M)/ZpId and PGL(Mn) := GL(Mn)/(Z/pn)Id, n ≥ 0. Eventually, denote by MrpM P(M), v 7→ v; MnrpMn P(Mn), vn 7→ vn; GL(M) PGL(M), γ 7→ γ;

and GL(Mn)PGL(Mn), γn7→γn the canonical projectivization morphisms.

LetGbe a closed subgroup of GL(M), which is automatically a compactp-adic analytic Lie group.

Consider an element v∈MrpM with the following property:

(*) For any open subgroupH ⊂G one has W =Qp[H]v.


Theorem 3.1. For any closed subgroupI ⊂G, one has:

(1) lim



|Gnvn| = 1

|I|; (2) lim



|Gnvn| = 1

|I|, where 1 := 0.

To prove theorem 3.1, we start with:


Theorem 3.2. For any closed subgroupJ ⊂G, one has:

(1) lim



|Gnvn| = 0, unless J is trivial;

(2) lim



|Gnvn| = 0, unless J is trivial.

Proof. We begin with a general lemma about projective limits of finite sets.

Lemma 3.3. Let A:= (An+1φn+1,n→ An)n≥0 be a projective system of nonempty finite sets such that (#) for alln≥0, there exists d(n)≥0, such that for all xn∈An, |φ−1n+1,n(xn)|=d(n) and let B := (Bn+1 φn+1,n→ Bn)n≥0 be a projective subsystem of A. Set B := lim

←− Bn and B∞,n :=

pn(B) ⊂ Bn, where pn : B → Bn is the canonical projection. In particular, (B∞,n+1 φn+1,n

→ B∞,n)n≥0 is a projective subsystem of B such that B= lim

←− B∞,n. Then (1) m(B) := lim



|An| and m(B) := lim



|An| exist and belong to [0,1].

(2) m(B) =m(B).

Proof of lemma 3.3. For (1), set δ(n) := max{|Bn+1∩φ−1n+1,n(xn)|}xn∈Bn ≤d(n). Then, by (#), one gets |B|An+1|



n|. Thus,m(B) exist and belongs to [0,1] since, by definition, |B|An|

n| ∈[0,1], n≥0. The same argument obviously works form(B).

For (2), the inequality m(B) ≤ m(B) follows from the inclusion B∞,n ⊂ Bn, n ≥ 0. For the opposite inequality, observe that for any n ≥ 0 there exists m(n) ≥ n such that φm(n),n(Bm(n)) = B∞,n. Indeed, else, set Bm,n := φ−1m,n(B∞,n) ⊂ Bm and B0m,n := Bm rBm,n for m ≥ n. Then (Bm+1,n0 → B0m,n)m≥n gives a well-defined projective subsystem of B. So, if B0m,n 6=∅ for all m≥ n, then

∅ 6= lim



B0m,n ⊂(lim


←−Bn,∞) =BrB=∅, which is a contradiction. Thus, one has

m(B)≤ |Bm(n)|

|Am(n)| ≤ δ(m(n)−1)· · ·δ(n) d(m(n)−1)· · ·d(n)


|An| ≤ |φm(n),n(Bm(n))|

|An| = |B∞,n|

|An| . Hence m(B)≤m(B).

Now, observe that A := (Gn+1vn+1 → Gnvn)n≥0 satisfies assumption (#). Indeed, for any xn = gnvn∈Gnvn and any xn+1=gn+1vn+1∈φ−1n+1,n(xn)⊂Gn+1vn+1 one has

−1n+1,n(xn)|= [StabGn+1(xn) : StabGn+1(xn+1)] = [StabGn+1(vn) : StabGn+1(vn+1)], which only depends on nbut not on xn∈Gnvn.1

So, to deduce (1) of theorem 3.2 from lemma 3.3, take B := ((Gn+1vn+1)J → (Gnvn)J)n≥0. Then B = (Gv)J and B∞,n = ((Gv)J)n, n ≥ 0. From lemma 3.3, it is enough to show that



|Gnvn| = 0. Denote by δ and δJ the dimension (as p-adic analytic space) of Gv and (Gv)J respectively. By [O82, Th. 2], there exist real (even rational) numbers µδ>0 and µδJ ≥0 such that

n→∞lim p−nδ|Gnvn|= µδ and lim

n→∞ p−nδJ|((Gv)J)n|=µδJ, respectively. (Observe that (Gv)n =Gnvn.) So, it is enough to prove that δJ < δ. Suppose otherwise, then there would exist an open subgroup

1Since GvM is a smooth closedp-adic analytic subspace, say, of dimensionδ, it even follows from [Se81, Rem. 1, p.148] that the canonical projectionsGn+1vn+1Gnvnarepδ-to-1 forn0.


H ⊂Gand an element g∈Gsuch thatgHv ⊂(Gv)J. But, from assumption (*), this implies thatJ acts trivially onW hence onM, as desired.

Similarly, to deduce (2) of theorem 3.2, define Bn ⊂ Gnvn, n ≥ 0, to be the inverse image of (Gnvn)J ⊂Gnvn under the canonical projectionGnvn Gnvn. Then B ={x∈Gv |γx=x, γ ∈ J} and, by lemma 3.3, it is enough to show that lim



|Gnvn| = 0. Since J is a compact p-adic Lie group, it is finitely generated. So, fixing a finite setγ(1), . . . , γ(r)of generators,B= \


B(i), where B(i):={x∈Gv|γ(i)x=x}. Now, B(i) is the finite disjoint union ofp-adic analytic closed subspaces indexed by the set spec(γ(i)) of eigenvalues of γ(i) inW:

B(i) = a


K(γ(i), λ),

where K(γ(i), λ) := {x ∈ Gv |γ(i)x = λx}. Since spec(γ(i)) is finite, theK(γ(i), λ) are also open in B(i). As in the proof of (1), by [O82, Th. 2], it is enough to prove that dim(B)< δ. Suppose other- wise, i.e., dim(B) =δ, then dim(B(i)) =δ must hold for all i= 1, . . . , r. Thus, for eachi= 1, . . . , r, there exists λ(i) ∈spec(γ(i)), such that K(γ(i), λ(i)) has dimension δ, hence contains an open subset gHv (for some open subgroupH⊂Gand some elementg∈G). But then assumption (*) yields that γ(i)(i)Idon W, hence onM. This means that J is trivial, as desired.

Proof of theorem 3.1. Given any finite groupF, write M(F) for the (finite) set of nontrivial minimal subgroups of F. Equivalently,M(F) is the set of cyclic subgroups ofF with prime order. (Note that M(F) =∅ if and only ifF ={1}.)

We first prove (1) by using (1) of theorem 3.2. Set (Gnvn)0 :=S

J∈M(In)(Gnvn)J, then one has 1




≤ |In\Gnvn|

|Gnvn| ≤ 1


1− |(Gnvn)0|



|Gnvn| . Since |In| → |I|, the only thing to prove is |(G|Gnvn)0|

nvn| →0.

IfI is finite, thenI is isomorphic to In forn0, so 0≤ |(Gnvn)0|

|Gnvn| ≤ X



|Gnvn| →0 (n→ ∞) by (1) of theorem 3.2.

IfI is infinite, thenI admits an infinite pro-pcyclic subgroup. Indeed, setKp:=Id+pi(p)Mm(Zp), where i(p) = 1 for p 6= 2 and i(2) = 2. Then the p-adic exponential defines an homeomorphism exp : pi(p)Mm(Zp) ' Kp such that exp(aM) = (exp(M))a, M ∈ pi(p)Mm(Zp), a ∈ Z. So, Kp is a torsion-free open normal pro-p subgroup of GLm(Zp) and I ∩Kp is an infinite pro-p torsion-free subgroup of GLm(Zp). Now, for any g∈(I∩Kp)r{Id}, one hashgi 'Zp, as desired.

So, up to replacing I by such a subgroup, one may assume I ' Zp. Fix any N ≥ 0 and set J =IpN ⊂I. Then, for anyxn∈Gnvnwith stabilizerIxn underI, note thatxn∈/ (Gnvn)J if and only ifJ *Ixn. But, since closed subgroups ofI 'Zp are totally ordered for⊂, the latter is also equivalent toIxn (J =IpN, or Ixn ⊂IpN+1 . So,xn∈/ (Gnvn)J implies |Ixn|= [I :Ixn]≥[I :IpN+1] = pN+1. Now,

0≤ |In\Gnvn|

|Gnvn| ≤ |In\(Gnvnr(Gnvn)J)|

|Gnvn| +|(Gnvn)J|

|Gnvn| ≤ 1

pN+1 +|(Gnvn)J|

|Gnvn| . By (1) of theorem 3.2, one has |(G|Gnvn)J|

nvn| →0, so 0≤ lim



|Gnvn| ≤ 1 pN+1,


which yields the desired conclusion as N ≥0 is arbitrary.

Finally, the proof for the projective situation (2) is exactly similar: use (2) (instead of (1)) of the- orem 3.2.

4. Proofs

4.1. Proof of theorem 1.1. 2 Denote the canonical epimorphismAη (Aη)0 =Aη/(Aη)0 by v 7→

v0. It induces a canonical epimorphism of Π-moduleAη[p](Aη)0[p] hence, for any v∈Aη[p] an inclusion of opens subgroups Πhvi ⊂Πhv0i ⊂Π corresponding to a commutative diagram of finite etale covers




S .

But then, by the Riemann-Hurwitz formula, Shvi has genus larger than Shv0i. The same observation obviously works for v, v0 as well instead of hvi, hv0i. So we can restrict to the case when (Aη)0 is trivial (that is,Aη contains no nontrivial abelian subvariety isogenous to an isotrivial abelian variety), which we assume from now on and till the end of this section.

In particular, to prove theorem 1.1, it is enough to prove the following.

Theorem 4.1. Assume that Aη contains no nontrivial abelian subvariety isogenous to an isotrivial abelian variety. Letv∈Tp(Aη) and setvn:=vmodpnTp(Aη)∈Aη[pn],n≥0. Then gvn → ∞ and ghvni→ ∞ (n→ ∞).

Indeed, assume that, say, the assersion forghviin theorem 1.1 does not hold. That is, there exists a c≥0 such that for alln≥0 there existsvn∈Aη[p] with ghvni≤cbutpnvn6= 0. From the inclusion of open subgroups Πhvni⊂Πhp−n|hvni|vni⊂Π, one getsghp−n|hvni|vni ≤ghvni ≤cso, one gets:

Aη[pn]c:={v∈Aη[pn] |ghvi≤c} 6=∅.

Furthermore, for any v ∈ Aη[pn]c the inclusion of open subgroups Πhvi ⊂ Πhpvi ⊂ Π yields, again, ghpvi≤ghvi. So{Aη[pn]c}n≥0forms a projective subsystem of nonempty (finite) subsets of (Aη[pn+1] → Aη[pn+1])n≥0 hence:

∅ 6= lim

←− Aη[pn]c ⊂lim

←− Aη[pn] =Tp(Aη), which contradicts theorem 4.1 forghvni. The case of gvn is just similar.

So, the rest of this section will be devoted to proving theorem 4.1.

4.1.1. The Π-module W(v). For anyv∈Tp(Aη), define:

W(v) :=Qp[Π]v⊂Vp(Aη).

Then W(v) contains the Qp-submodule:

W(v) := \



where the intersection is over all open subgroups H ⊂ Π. By definition v ∈ W(v) so W(v) is nonzero. Also, one can write W(v) = Qp[H0]v for some open subgroup H0 ⊂ Π. Indeed, for any open subgroups H1, H2⊂Π observe thatQp[H1]v∩Qp[H2]v⊃Qp[H1∩H2]v. So any open subgroup H0⊂Π such thatQp[H0]v has minimal Qp-dimension works.

In particular, the closed subgroup

Π(v) :={γ ∈Π|γ·W(v) =W(v)} ⊂Π

2Asgvghvi, the assertion forgvis implied by the one forghvi. However, here we treat both cases in parallel, partly because the proof forgvis somewhat simpler than the one forghvi, and the former may help the readers to understand the latter.


containsH0 hence is also open in Π.

Furthermore, from the inclusion:



Πhvni= Πhvi ⊂Π(v), one obtains:

Π = Π(v)[ [



So it follows from the compactness of Π that there exists an integerN ≥0 such that:

Π = Π(v)[ [


(ΠrΠhvni), hence:

ΠhvNi= \



In particular Πhvni⊂Π(v), n0. Similarly, Πvn ⊂Π(v),n0.

But, since we are interested in estimating gvn, ghvni for n 0, one may replace Π by Π(v) and W(v) byW(v). So, from now on we assume that:

(*) For any open subgroup H⊂Π, one has W(v) =Qp[H]v.

4.1.2. The full level-pn structure S(n)→S. SetM :=W(v)∩Tp(Aη)('Zmp ). We retain the notation of section 3 and denote byρ: Π→GL(M) the corresponding representation, and byG⊂GL(M) its image. Then G is a compact subgroup of GLm(Zp). So it is an almost-pro-p compact p-adic infinite (cf. corollary 2.4) Lie-group.

Denote by Π(n) C Π and Gn 'Π/Π(n) ⊂ GL(Mn) the kernel and the image of Π→ρ GL(M) GL(Mn) respectively. Finally, set G(n) := ρ(Π(n)) ⊂G and G# := ρ(Π#) (# = v, hvi, etc). The inclusion of open subgroups Π(n)⊂Πvn ⊂Πhvni⊂Π corresponds to the sequence of finite etale covers

S(n)→Svn →Shvni →S

whereS(n)→S is Galois with groupGn. Writeg(n) for the genus of the smooth compactification of S(n).

4.1.3. g(n) → ∞. As already noticed, |Gn| → ∞ by corollary 2.4, so, by the Riemann-Hurwitz for- mula, sup{g(n)}<∞ if and only if sup{g(n)} ≤1.

If sup{g(n)} = 1, then there exists n0 ≥ 0 such that g(n) = 1 for n ≥ n0. Then the smooth compactification S(n0) of S(n0) is an elliptic curve and thatG(n0) = lim

←− (Π(n0)/Π(n)) is a quotient of π1(S(n0)) (Zˆ2), which contradicts corollary 2.4.

If sup{g(n)} = 0, then S(n) → S is a Galois cover of genus 0 curve with degree |Gn| → ∞. Set i(p) = 1 for p6= 2 and i(2) = 2. Then, replacing S by S(i(p)) if necessary, we may assume: (i) Gn is a p-group; and (ii) any element of Gn=G/G(n) of order p is contained in G(n−1)/G(n), hence in the center of Gn. Indeed, (i) is clear and (ii) follows from a standard argument involving the p-adic exponential, as in the proof of theorem 3.1. Now, we resort to the classification of finite subgroups of PGL2(k) = Aut(P1k). More specifically, it follows from [Su82, Th. 6.17, Case I] and condition (i) above that eitherGn is a cyclicp-group orp= 2 andGn is a dihedral group of order 2m+1. Moreover, condition (ii) above forcesm∈ {0,1} in the latter case. So, as|Gn| → ∞, (Gn+1 →Gn)n≥0 must be a projective system of cyclic p-groups. Thus,G is abelian, which contradicts corollary 2.4.

Remark 4.2. When k = C and M is the whole p-adic Tate module Tp(Aη) (with Aη principally polarized), J.-M. Hwang and W.-K. To proved that a uniform bound (i.e., depending only on dim(Aη)) for the growth ofg(n) exists [HT06]. By classical arguments (Zarhin’s trick and specialization), such a uniform bound for Tp(Aη) also exists only under the assumption that khas characteristic 0.


4.1.4. gvn → ∞. First, for any connected finite etale cover T → S of degree d, set λ(T → S) :=


d , whereg(T) stands for the genus of the smooth compactification ofT. Then, ifU →T →S are connected finite etale covers, it follows from the Riemann-Hurwitz formula for U → T that λ(U)≥λ(T).

Now, set λn:= λ(S(n) →S) and λvn :=λ(Svn → S). Thus, one has λn+1 ≥λnvn+1 ≥λvn and λn≥λvn,n≥0. In particular,λ:= lim

n→∞λn and λv:= lim

n→∞λvn exist andλ≥λv.

As already noticed, |Gn| → ∞ (by corollary 2.4), and, as well, |Gnvn| → ∞. Indeed, else, since Gnvn ' G/Gvn, Πv would be a closed subgroup of finite index in Π, hence open in Π. Thus, by assumption (*),W(v) =Qpv]v=Qpvis 1-dimensional, which contradicts corollary 2.4. So,gn→ ∞ if and only ifλ >0, andgvn → ∞ if and only ifλv >0. But from paragraph 4.1.3,λ >0 holds. Thus, it is enough to prove that λ=λv.

To do this, we shall rewriteλnand λvn in group-theoretic terms, by means of the Riemann-Hurwitz formula. More specifically, for i = 1, ..., r, write Ii,n ⊂ Gn for the image of the inertia group at Pi in Gn and denote by di,n and ei,n (resp. dn(P) and en(P)) the exponent of the different and the ramification index of any place of S(n) (resp. of the placeP of Svn) above Pi in S(n)→S (resp. in Svn →S). Eventually, for any placeP ofSvn, writedn(P) anden(P) for the exponent of the different and the ramification index of any place of S(n) above P inS(n) → Svn. Recall that these data are linked by the relations

ei,n=en(P)en(P), di,n=en(P)dn(P) +dn(P).

Now, by the Riemann-Hurwitz formula, one has:

λn:= 2g(0)−2 + X


di,n ei,n

, λvn = 2g(0)−2 + 1

|Gnvn| X



P∈Svn, P|Pi


Setn:=λn−λvn ≥0. Then observe n= X


i,n, i,n:= 1



P∈Svn, P|Pi

dn(P) en(P). Moreover, it is classically known ([Se68, Chap. IV, Prop. 4]) that

i,n= 1



P∈Svn, P|Pi



en(P)j−1 en(P) ,

whereen(P)j denotes the order of thejth ramification groupIn(P)j (with “lower numbering”) of any place ofS(n) aboveP inS(n)→Svn,j≥0. (Thus, en(P)0 =en(P), and en(P)j = 1 for j0.)

AsG(1)/G(n) =ker(Gn→G1) is ap-group, so is Ii,n(1) :=Ii,n∩(G(1)/G(n)) =Ker(Ii,n→Ii,1).

Thus, (sincep6=char(k)) the natural surjectionIi,n→Ii,1 induces an isomorphism (Ii,n)+→(Ii,1)+, where (Ii,n)+denotes the wild inertia subgroup ofIi,n, i.e., (Ii,n)+is the trivial (resp. uniqueq-Sylow) subgroup of Ii,n if the characteristic of k is 0 (resp. q). Now, for integers j ≥ 0 and n ≥ 0, define a subgroup ( ˜Ii,n)j of Ii,n as follows: ( ˜Ii,n)0 := Ii,n and, for j > 0, ( ˜Ii,n)j is the inverse image of the jth ramification group (Ii,1)j of Ii,1 under the above isomorphism (Ii,n)+ → (Ii,1)+. (Observe, in particular, that there existsj0≥0 such that ( ˜Ii,n)j ={1} forj > j0.) Forj >0, thejth ramification group (Ii,n)j of Ii,nmaps surjectively onto (Ii,1)dj/ei,n(1)eby the above isomorphism (Ii,n)+→(Ii,1)+, where ei,n(1) :=|Ii,n(1)| ([Se68, Chap. IV, Lem. 5]). From this, one obtains (Ii,n)j = ( ˜Ii,n)dj/ei,n(1)e forj≥0. (Note that this is clear forj = 0.)

Moreover, define ( ˜Ii)j to be the projective limit of {( ˜Ii,n)j}n≥0. Thus, ( ˜Ii)0 coincides with the inertia subgroup Ii ⊂ G at Pi, while, for j > 0, ( ˜Ii)j is a finite, normal q-subgroup of Ii which is projected isomorphically onto ( ˜Ii,n)j for each n >0.

Lemma 4.3. Let I be a finite group and J a subgroup of I. Let X be a finite set on which I acts.

Consider the natural surjection of quotient sets: J\X → I\X. Then, for any x ∈ X, the fiber of


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