SCHEMES (P reliminary version)

ANNA CADORET AND AKIO TAMAGAWA

Abstract. ^{Let}^{k}be a field finitely generated overQand denote by Γk its absolute Galois group. LetAbe
an abelian variety overk. Given a primepand a characterχ: Γ_{k}→Z^{∗}p, define thetwistedp-primary torsion
associated withχto be the Γ_{k}-module:

A[p^{∞}](χ) :={T ∈A[p^{∞}]|^{σ}T =χ(σ)T , σ∈Γk},

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[p^{N}] holds for anys∈S(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 anyc≥0, there exists an integer
N:=N(c, A, S, k, p)≥0 such thatA[p^{∞}](K^{0})⊂A[p^{N}] for all finite extensionK^{0}/KwithK^{0}of genus≤c.

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+1→Hn) 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+1→Gn) 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 A_{η} →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 Γ_{η} on the Tate module T_{p}(A_{η}) factors through Π. This, in turn, defines an action
of Π overAη[p^{n}],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 Π_{hvi} for the stabilizer of hvi under Π and S_{hvi} → S for the resulting connected finite
etale cover (defined over a finite extension k_{hvi}/k). The inclusion of open subgroups Π_{v} ⊂Π_{hvi} ⊂ Π

1

yields a commutative diagram of finite etale covers:

S_{v} ^{//}

S_{hvi}

}}{{{{{{{{

S

We will write gv and ghvi for the genus of the smooth compactifications of Sv×_{k}_{v}k and Shvi×_{k}_{hvi}k,
respectively.

Eventually, denote by (A_{η})_{0} the largest abelian subvariety ofA_{η} 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 ∈ A_{η}[p^{∞}] either g_{v} ≥ c or p^{N}v ∈ (A_{η})_{0}. The same
statement holds if gv is replaced by g_{hvi}.

Say that a character χ : Γk → Z^{∗}_{p} 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 → Z^{∗}_{p} 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 A_{s}[p^{∞}](χ) :={T ∈A_{s}[p^{∞}]|^{σ}T =χ(σ)T, σ∈Γ_{k}} is contained inA_{s}[p^{N}].

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 modp^{n}Tp(Aη)∈Aη[p^{n}]^{∗},
n≥0. Then g_{v}_{n} → ∞ and g_{hv}_{n}_{i} → ∞.

Then, to any v ∈ T_{p}(A_{η})^{∗} associate the Π-module M := Qp[Π]v∩T_{p}(A_{η}) and denote by Π(n)
the kernel of the reduction morphism GL(M) GL(M/p^{n}). The inclusion of open subgroups
Π(n) ⊂Π_{v}_{n} ⊂ Π corresponds to a sequence of finite etale covers S(n) → S_{v}_{n} → 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” ofg_{v}_{n} 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 modulop^{n} of p-adic analytic subspaces of
Z^{m}p . The proof of the variant forg_{hvi} 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
coversS_{n+1,χ}→ S_{n,χ}ofS with the property thatk-rational pointss_{n}:k→ S_{n,χ}lying aboves:k→S
correspond to elements of order exactly p^{n} inAs[p^{∞}](χ). It is thus enough to prove that S_{n,χ}(k) =∅
for n 0. According to theorem 1.1, S_{n,χ} can be written as a disjoint union S_{n,χ} = S_{n,χ}^{(1)}`

S_{n,χ}^{(2)},
where S_{n,χ}^{(1)} corresponds to the “isotrivial part” of A_{η}[p^{n}] and S_{n,χ}^{(2)} to the “non-isotrivial” one. On
one hand, we show that S_{n,χ}^{(1)}(k) =∅ forn0, and, on the other hand, from theorem 1.1, the genus
of each component of S_{n,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

(H_{n+1}→H_{n}) 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 (G_{n+1}→G_{n}) 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 G_{n} the dihedral groupD_{2p}^{n} of order 2p^{n}, (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 p^{n}. In particular, for s = 2, (H_{n+1} → H_{n}) is just the
projective system of modular curves (Y1(p^{n+1})→Y1(p^{n})) 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(p^{n})(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 H_{n} 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
Pic^{0}_{C}

0|H0 →H_{0} of the universal curve C_{0} →H_{0}. WhenH_{n} 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 ˜H_{n}(k)6=∅ as soon asH_{n}(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 modulop^{n} 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[p^{n}] :=ker([p^{n}] :A(k)→A(k)) for the kernel of the multiplication-by-p^{n} endomorphism;

A[p^{∞}] := [

n≥0

A[p^{n}] for thep-primary torsion ofA(k).

T_{p}(A) := lim

←− A[p^{n}] for thep-adic Tate module ofA;

A[p^{n}]^{∗}:=A[p^{n}]rpA[p^{n}];

Tp(A)^{∗} :=Tp(A)rpTp(A) = lim

←− A[p^{n}]^{∗};
Vp(A) :=Tp(A)⊗_{Z}_{p}Qp;

Now, if A → η is defined over the function field η of a k-curve S → k, we will write A_{0} 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×_{η}η descends to an abelian variety over k.) Equivalently,
A_{0} 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,A_{2} is isogeneous to an abelian subvariety ofA_{1}. 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 ofA_{1}. In particular, ifA_{i}⊂Ais isogeneous to an isotrivial abelian variety,i= 1,2,
then so is A_{1}+A_{2}⊂A (as a homomorphic image ofA_{1}×A_{2}); thus A_{0} 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 A^{0} to be the quotient abelian variety
A/A_{0}. Then (again by Poincar´e’s complete reducibility theorem) we have (A^{0})_{0}= 0.

2.2. Non-Tate characters. Fix a prime p and let k be a field of characteristic q 6= p. Let χ :
Γ_{k} →Z^{∗}p 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 k^{0} of k, χ : Γ_{k} → Z^{∗}_{p} is non-Tate if and only if
χ|_{Γ}

k0 : Γk^{0} →Z^{∗}_{p} is non-Tate.

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

k0 appears in Tp(A×_{k}k^{0}). Next, to show the ‘only if’ part, suppose that χ|_{Γ}

k0 appears in Tp(A^{0})
for some abelian varietyA^{0} overk^{0}.

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

HomΓ_{k}(Zp(χ), Tp(A)) = lim

←−Homk(Mn, A) = lim

←−Hom_{k}^{0}(Mn×_{k}k^{0}, A^{0}) = HomΓ_{k}0(Zp(χ|_{Γ}

k0), Tp(A^{0})).

In the latter case, take N ≥0 such that (k^{0})^{q}^{N} ⊂k and set A:= A^{0}×_{k}^{0}k, where k^{0} → k is given by
a7→ a^{q}^{N}. Then, by using the fact that the q^{N}th power map k → k induces the identity on Γ_{k}, one
can show thatχ appears inT_{p}(A).

Finally, assume that k^{0} is an arbitrary finitely generated extension of k and that χ|_{Γ}

k0 appears in
Tp(A^{0}) for some abelian variety A^{0} over k^{0}. Then, taking a model of A^{0} → k^{0} over the spectrum of
a finitely generated k-subalgebra of k^{0} whose fraction field coincides with k^{0} and specializing it at a
closed point, one can show that there exist a finite extension k^{00} of k and an abelian variety A^{00} over
k^{00} such thatχ|_{Γ}

k00 appears inT_{p}(A^{00}). Thus, by the preceding argument,χappears inT_{p}(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 Aut_{Q}_{p}(V) (e.g., trivial, abelian, unipotent, pro-p,
etc.), say that a subgroupGof Aut_{Q}_{p}(V) isalmost-P if there exists a closed subgroupH ⊂Gof finite
index such thatH hasP. Further, given a group Π and a representationρ: Π→Aut_{Q}_{p}(V), say that
ρ : Π → Aut_{Q}_{p}(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:π_{1}(S)→Aut_{Z}_{p}(T_{p}(A_{η}))⊂Aut_{Q}_{p}(V_{p}(A_{η})).

Proposition 2.2. Let M be a π_{1}(S)-submodule of T_{p}(A_{η}) and and consider the corresponding repre-
sentation ρM :π1(S)→Aut_{Z}_{p}(M)⊂Aut_{Q}_{p}(M⊗_{Z}_{p}Qp). Then the following are all equivalent.

(1) ρ_{M} is almost-trivial (i.e., ρ_{M} has finite image).

(2) ρM is almost-abelian.

(3) ρ_{M} is almost-unipotent.

(4) M ⊂T_{p}((A_{η})_{0}).

Proof. (1)⇒(2). Clear.

(2)⇒(3). First observe Aut_{Z}_{p}(M)'GL_{m}(Zp), wheremis the rank of the freeZp-moduleM. Thus,
from the short exact sequence

1→1 +pM_{m}(Zp)→GL_{m}(Zp)→GL_{m}(Z/p)→1,

the group Aut_{Z}_{p}(M) is almost-pro-p. This, together with the assumption that ρ_{M} is almost-abelian,
assures that one may assume that ρM(π1(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 k_{1} 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 k_{1} and A_{1} → S_{1} 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 π_{1}(S) π_{1}((S_{1})_{k}

1) ([SGA1, Exp. X, proof of Cor. 1.6]). At the level of Tate
modules, one has a canonical isomorphism T_{p}(A_{η}) ˜→T_{p}(A_{1,η}_{1}) 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 π_{1}(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→Γ_{k}_{1} →1.

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

Lemma 2.3. For any finite extensionk^{0}_{1} of k1, the coinvariant quotient ∆Γ_{k}0
1

is finite.

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

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

Now, define ∆ and Π to be the images of ∆ and Π in Aut_{Z}_{p}(M). SetR_{0}:=Zp[∆]⊂End_{Z}_{p}(M) and
R := R0 ⊗_{Z}_{p} Qp. As R0 is a free Zp-module, the canonical morphism R0 → R is injective, hence so
is R^{red}_{0} →R^{red}, where R^{red}_{0} =R_{0}/√

0_{R}_{0} and R^{red} :=R/√

0_{R}. We shall denote the natural surjection
R R^{red} by f 7→ f^{red}. 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 R^{red}. Denote
by i: Π Π → Aut_{Q}_{p}_{-alg}(R) and i^{red}: Π →Aut_{Q}_{p}_{-alg}(R^{red}) the resulting representations. Observe
thati (hence alsoi^{red}) factors through ΠΓk1, since ∆ is abelian.

Since R^{red} is a finite reduced commutativeQp-algebra, it is isomorphic to a product Q

1≤i≤rLi of
finite field extensionsL_{i}/Qp,i= 1, ..., rand itsQp-algebra automorphism group is finite. In particular,
i^{red}has finite image. So, up to replacingk1 by a finite extension, one may assume thati^{red} is trivial.

Then, if we denote the image of ∆ in (R^{red}_{0} )^{∗} ⊂(R^{red})^{∗} by ∆^{red}, it follows that the natural surjection

∆∆^{red}factors through ∆_{Γ}_{k}

1. Thus, by lemma 2.3, ∆^{red}is finite.

Now, up to replacing S by a finite etale cover, ∆ has trivial image in (R^{red})^{∗}, or, equivalently, has
unipotent image in R^{∗}⊂Aut(M⊗_{Z}_{p}Qp), as desired.

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

(4)⇒(1). Immediate from the definition of (A_{η})0.
Corollary 2.4. The following are all equivalent.

(1) T_{p}(A_{η}) admits no nonzeroπ_{1}(S)-submoduleM withρ_{M} almost-trivial.

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

(3) T_{p}(A_{η}) admits no nonzeroπ_{1}(S)-submoduleM withρ_{M} 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 ⊂T_{p}((A_{η})_{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= Pic^{0}_{X/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 p^{n} of p-adic analytic homogeneous spaces

The aim of this section is to prove theorem 3.1 concerning reduction modulo p^{n} 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/p^{n}, n ≥ 0,
and W := M ⊗_{Z}_{p} Qp. Set GL(M) := Aut_{Z}_{p}(M) and GL(M_{n}) := Aut_{Z/p}^{n}(M_{n}), n ≥ 0. Let M
M_{n}, v 7→v_{n} denote reduction modulo p^{n}M and GL(M)GL(M_{n}), g 7→g_{n} the induced morphism.

As projective versions of these, set P(M) := (M rpM)/Z^{∗}p and P(Mn) := (MnrpMn)/(Z/p^{n})^{∗},
n ≥ 0. Set PGL(M) := GL(M)/Z^{∗}pId and PGL(Mn) := GL(Mn)/(Z/p^{n})^{∗}Id, n ≥ 0. Eventually,
denote by MrpM P(M), v 7→ v; M_{n}rpM_{n} P(M_{n}), v_{n} 7→ v_{n}; GL(M) PGL(M), γ 7→ γ;

and GL(Mn)PGL(Mn), γn7→γ_{n} 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.

Then:

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

(1) lim

n→∞

|I_{n}\G_{n}v_{n}|

|G_{n}vn| = 1

|I|; (2) lim

n→∞

|I_{n}\G_{n}vn|

|G_{n}vn| = 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

n→∞

|(G_{n}v_{n})^{J}|

|G_{n}vn| = 0, unless J is trivial;

(2) lim

n→∞

|(G_{n}v_{n})^{J}|

|G_{n}vn| = 0, unless J is trivial.

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

Lemma 3.3. Let A:= (A_{n+1}^{φ}^{n+1,n}→ A_{n})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^{−1}_{n+1,n}(xn)|=d(n)
and let B := (B_{n+1} ^{φ}^{n+1,n}→ B_{n})n≥0 be a projective subsystem of A. Set B∞ := lim

←− B_{n} and B∞,n :=

p_{n}(B∞) ⊂ B_{n}, where p_{n} : B∞ → B_{n} 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

n→∞

|B_{n}|

|A_{n}| and m∞(B) := lim

n→∞

|B_{∞,n}|

|A_{n}| exist and belong to [0,1].

(2) m(B) =m∞(B).

Proof of lemma 3.3. For (1), set δ(n) := max{|B_{n+1}∩φ^{−1}_{n+1,n}(x_{n})|}_{x}_{n}_{∈B}_{n} ≤d(n). Then, by (#), one
gets ^{|B}_{|A}^{n+1}^{|}

n+1| ≤ ^{δ(n)}_{d(n)}^{|B}_{|A}^{n}^{|}

n| ≤ ^{|B}_{|A}^{n}^{|}

n|. Thus,m(B) exist and belongs to [0,1] since, by definition, ^{|B}_{|A}^{n}^{|}

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 φ_{m(n),n}(B_{m(n)}) =
B∞,n. Indeed, else, set B_{m,n} := φ^{−1}_{m,n}(B∞,n) ⊂ Bm and B^{0}_{m,n} := Bm rB_{m,n} for m ≥ n. Then
(B_{m+1,n}^{0} → B^{0}_{m,n})m≥n gives a well-defined projective subsystem of B. So, if B^{0}_{m,n} 6=∅ for all m≥ n,
then

∅ 6= lim

←−

m

B^{0}_{m,n} ⊂(lim

←−B_{n})r(lim

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

m(B)≤ |B_{m(n)}|

|A_{m(n)}| ≤ δ(m(n)−1)· · ·δ(n)
d(m(n)−1)· · ·d(n)

|φ_{m(n),n}(B_{m(n)})|

|A_{n}| ≤ |φ_{m(n),n}(B_{m(n)})|

|A_{n}| = |B∞,n|

|A_{n}| .
Hence m(B)≤m∞(B).

Now, observe that A := (G_{n+1}v_{n+1} → G_{n}v_{n})n≥0 satisfies assumption (#). Indeed, for any x_{n} =
g_{n}v_{n}∈G_{n}v_{n} and any x_{n+1}=g_{n+1}v_{n+1}∈φ^{−1}_{n+1,n}(x_{n})⊂G_{n+1}v_{n+1} one has

|φ^{−1}_{n+1,n}(xn)|= [StabGn+1(xn) : StabGn+1(xn+1)] = [StabGn+1(vn) : StabGn+1(vn+1)],
which only depends on nbut not on x_{n}∈G_{n}v_{n}.^{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

n→∞lim

|((Gv)^{J})_{n}|

|G_{n}vn| = 0. Denote by δ and δ_{J} the dimension (as p-adic analytic space) of Gv and (Gv)^{J}
respectively. By [O82, Th. 2], there exist real (even rational) numbers µ_{δ}>0 and µ_{δ}_{J} ≥0 such that

n→∞lim p^{−nδ}|G_{n}vn|= µδ and lim

n→∞ p^{−nδ}^{J}|((Gv)^{J})n|=µδ_{J}, respectively. (Observe that (Gv)n =Gnvn.)
So, it is enough to prove that δ_{J} < δ. Suppose otherwise, then there would exist an open subgroup

1Since Gv⊂M is a smooth closedp-adic analytic subspace, say, of dimensionδ, it even follows from [Se81, Rem. 1,
p.148] that the canonical projectionsGn+1vn+1→Gnvnarep^{δ}-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 B_{n} ⊂ G_{n}v_{n}, 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

n→∞

|B_{∞,n}|

|G_{n}v_{n}| = 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∞= \

1≤i≤r

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

λ∈spec(γ^{(i)})

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)}=λ^{(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 (G_{n}v_{n})^{0} :=S

J∈M(I_{n})(G_{n}v_{n})^{J}, then one has
1

|I_{n}|

1−|(G_{n}v_{n})^{0}|

|G_{n}vn|

≤ |I_{n}\G_{n}v_{n}|

|G_{n}vn| ≤ 1

|I_{n}|

1− |(G_{n}v_{n})^{0}|

|G_{n}vn|

+|(G_{n}v_{n})^{0}|

|G_{n}vn| .
Since |I_{n}| → |I|, the only thing to prove is ^{|(G}_{|G}^{n}^{v}^{n}^{)}^{0}^{|}

nvn| →0.

IfI is finite, thenI is isomorphic to In forn0, so
0≤ |(G_{n}vn)^{0}|

|G_{n}v_{n}| ≤ X

J∈M(I)

|(G_{n}vn)^{J}|

|G_{n}v_{n}| →0 (n→ ∞)
by (1) of theorem 3.2.

IfI is infinite, thenI admits an infinite pro-pcyclic subgroup. Indeed, setKp:=Id+p^{i(p)}Mm(Zp),
where i(p) = 1 for p 6= 2 and i(2) = 2. Then the p-adic exponential defines an homeomorphism
exp : p^{i(p)}M_{m}(Zp) ' K_{p} such that exp(aM) = (exp(M))^{a}, M ∈ p^{i(p)}M_{m}(Zp), a ∈ Z. So, K_{p} is
a torsion-free open normal pro-p subgroup of GLm(Zp) and I ∩Kp is an infinite pro-p torsion-free
subgroup of GL_{m}(Zp). Now, for any g∈(I∩K_{p})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 =I^{p}^{N} ⊂I. Then, for anyx_{n}∈G_{n}v_{n}with stabilizerI_{x}_{n} underI, note thatx_{n}∈/ (G_{n}v_{n})^{J} if and only
ifJ *Ixn. But, since closed subgroups ofI 'Zp are totally ordered for⊂, the latter is also equivalent
toI_{x}_{n} (J =I^{p}^{N}, or I_{x}_{n} ⊂I^{p}^{N+1} . So,x_{n}∈/ (G_{n}v_{n})^{J} implies |Ix_{n}|= [I :I_{x}_{n}]≥[I :I^{p}^{N+1}] = p^{N+1}.
Now,

0≤ |I_{n}\G_{n}v_{n}|

|G_{n}v_{n}| ≤ |I_{n}\(G_{n}v_{n}r(G_{n}v_{n})^{J})|

|G_{n}v_{n}| +|(G_{n}v_{n})^{J}|

|G_{n}v_{n}| ≤ 1

p^{N}^{+1} +|(G_{n}v_{n})^{J}|

|G_{n}v_{n}| .
By (1) of theorem 3.2, one has ^{|(G}_{|G}^{n}^{v}^{n}^{)}^{J}^{|}

nvn| →0, so 0≤ lim

n→∞

|I_{n}\G_{n}v_{n}|

|G_{n}v_{n}| ≤ 1
p^{N}^{+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 epimorphismA_{η} (A_{η})^{0} =A_{η}/(A_{η})_{0} by v 7→

v^{0}. It induces a canonical epimorphism of Π-moduleA_{η}[p^{∞}](A_{η})^{0}[p^{∞}] hence, for any v∈A_{η}[p^{∞}]
an inclusion of opens subgroups Πhvi ⊂Π_{hv}^{0}_{i} ⊂Π corresponding to a commutative diagram of finite
etale covers

S_{hvi}

//S_{hv}^{0}_{i}

||xxxxxxxx

S .

But then, by the Riemann-Hurwitz formula, Shvi has genus larger than S_{hv}^{0}_{i}. The same observation
obviously works for v, v^{0} as well instead of hvi, hv^{0}i. So we can restrict to the case when (A_{η})_{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 A_{η} contains no nontrivial abelian subvariety isogenous to an isotrivial
abelian variety. Letv∈T_{p}(A_{η})^{∗} and setv_{n}:=vmodp^{n}T_{p}(A_{η})∈A_{η}[p^{n}]^{∗},n≥0. Then g_{v}_{n} → ∞ and
g_{hv}_{n}_{i}→ ∞ (n→ ∞).

Indeed, assume that, say, the assersion forg_{hvi}in theorem 1.1 does not hold. That is, there exists a
c≥0 such that for alln≥0 there existsv_{n}∈A_{η}[p^{∞}] with g_{hv}_{n}_{i}≤cbutp^{n}v_{n}6= 0. From the inclusion
of open subgroups Π_{hv}_{n}_{i}⊂Π_{hp}^{−n}_{|hv}_{n}_{i|v}_{n}_{i}⊂Π, one getsg_{hp}^{−n}_{|hv}_{n}_{i|v}_{n}_{i} ≤g_{hv}_{n}_{i} ≤cso, one gets:

A_{η}[p^{n}]^{∗}_{c}:={v∈A_{η}[p^{n}]^{∗} |g_{hvi}≤c} 6=∅.

Furthermore, for any v ∈ Aη[p^{n}]^{∗}_{c} the inclusion of open subgroups Π_{hvi} ⊂ Π_{hpvi} ⊂ Π yields, again,
g_{hpvi}≤g_{hvi}. So{A_{η}[p^{n}]^{∗}_{c}}_{n≥0}forms a projective subsystem of nonempty (finite) subsets of (A_{η}[p^{n+1}]^{∗} →
Aη[p^{n+1}]^{∗})n≥0 hence:

∅ 6= lim

←− Aη[p^{n}]^{∗}_{c} ⊂lim

←− Aη[p^{n}]^{∗} =Tp(Aη)^{∗},
which contradicts theorem 4.1 forg_{hv}_{n}_{i}. 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) := \

H⊂Π

Qp[H]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[H_{0}]v for some open subgroup H_{0} ⊂ Π. Indeed, for any
open subgroups H_{1}, H_{2}⊂Π observe thatQp[H_{1}]v∩Qp[H_{2}]v⊃Qp[H_{1}∩H_{2}]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

2Asgv≥ghvi, 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.

containsH_{0} hence is also open in Π.

Furthermore, from the inclusion:

\

n≥0

Π_{hv}_{n}_{i}= Π_{hvi} ⊂Π∞(v),
one obtains:

Π = Π∞(v)[ [

n≥0

(ΠrΠ_{hv}_{n}_{i}).

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

Π = Π∞(v)[ [

n≤N

(ΠrΠ_{hv}_{n}_{i}),
hence:

Π_{hv}_{N}_{i}= \

n≤N

Π_{hv}_{n}_{i}⊂Π∞(v).

In particular Πhv_{n}i⊂Π∞(v), n0. Similarly, Πvn ⊂Π∞(v),n0.

But, since we are interested in estimating g_{v}_{n}, g_{hv}_{n}_{i} 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-p^{n} structure S(n)→S. SetM :=W(v)∩T_{p}(A_{η})('Z^{m}p ). 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(M_{n}) respectively. Finally, set G(n) := ρ(Π(n)) ⊂G and G_{#} := ρ(Π_{#}) (# = v, hvi, etc). The
inclusion of open subgroups Π(n)⊂Π_{v}_{n} ⊂Π_{hv}_{n}_{i}⊂Π corresponds to the sequence of finite etale covers

S(n)→S_{v}_{n} →S_{hv}_{n}_{i} →S

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

4.1.3. g(n) → ∞. As already noticed, |G_{n}| → ∞ 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 n_{0} ≥ 0 such that g(n) = 1 for n ≥ n_{0}. Then the smooth
compactification S(n_{0}) of S(n_{0}) is an elliptic curve and thatG(n_{0}) = lim

←− (Π(n_{0})/Π(n)) is a quotient
of π1(S(n0)) (Zˆ^{2}), which contradicts corollary 2.4.

If sup{g(n)} = 0, then S(n) → S is a Galois cover of genus 0 curve with degree |G_{n}| → ∞. 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 G_{n}=G/G(n) of order p is contained in G(n−1)/G(n), hence in
the center of G_{n}. 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 PGL_{2}(k) = Aut(P^{1}_{k}). More specifically, it follows from [Su82, Th. 6.17, Case I] and condition (i)
above that eitherG_{n} is a cyclicp-group orp= 2 andG_{n} is a dihedral group of order 2^{m+1}. Moreover,
condition (ii) above forcesm∈ {0,1} in the latter case. So, as|G_{n}| → ∞, (G_{n+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 T_{p}(A_{η}) (with A_{η} 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. g_{v}_{n} → ∞. First, for any connected finite etale cover T → S of degree d, set λ(T → S) :=

2g(T)−2

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 λ_{n}:= λ(S(n) →S) and λ_{v}_{n} :=λ(S_{v}_{n} → S). Thus, one has λ_{n+1} ≥λ_{n},λ_{v}_{n+1} ≥λ_{v}_{n} and
λn≥λvn,n≥0. In particular,λ:= lim

n→∞λn and λv:= lim

n→∞λvn exist andλ≥λv.

As already noticed, |G_{n}| → ∞ (by corollary 2.4), and, as well, |G_{n}vn| → ∞. Indeed, else, since
G_{n}v_{n} ' G/G_{v}_{n}, Π_{v} would be a closed subgroup of finite index in Π, hence open in Π. Thus, by
assumption (*),W(v) =Qp[Πv]v=Qpvis 1-dimensional, which contradicts corollary 2.4. So,gn→ ∞
if and only ifλ >0, andg_{v}_{n} → ∞ if and only ifλ_{v} >0. But from paragraph 4.1.3,λ >0 holds. Thus,
it is enough to prove that λ=λv.

To do this, we shall rewriteλ_{n}and λ_{v}_{n} in group-theoretic terms, by means of the Riemann-Hurwitz
formula. More specifically, for i = 1, ..., r, write I_{i,n} ⊂ G_{n} for the image of the inertia group at P_{i}
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 S_{v}_{n}) above P_{i} in S(n)→S (resp. in
Svn →S). Eventually, for any placeP ofSvn, writed^{n}(P) ande^{n}(P) for the exponent of the different
and the ramification index of any place of S(n) above P inS(n) → S_{v}_{n}. Recall that these data are
linked by the relations

ei,n=e^{n}(P)en(P), di,n=e^{n}(P)dn(P) +d^{n}(P).

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

λn:= 2g(0)−2 + X

1≤i≤r

d_{i,n}
ei,n

, λvn = 2g(0)−2 + 1

|G_{n}vn|
X

1≤i≤r

X

P∈S_{vn}, P|Pi

dn(P).

Set_{n}:=λ_{n}−λ_{v}_{n} ≥0. Then observe
n= X

1≤i≤r

i,n, i,n:= 1

|G_{n}v_{n}|

X

P∈S_{vn}, P|Pi

d^{n}(P)
e^{n}(P).
Moreover, it is classically known ([Se68, Chap. IV, Prop. 4]) that

_{i,n}= 1

|G_{n}v_{n}|

X

P∈S_{vn}, P|P_{i}

X

j≥0

e^{n}(P)_{j}−1
e^{n}(P) ,

wheree^{n}(P)j denotes the order of thejth ramification groupIn(P)j (with “lower numbering”) of any
place ofS(n) aboveP inS(n)→S_{v}_{n},j≥0. (Thus, e^{n}(P)_{0} =e^{n}(P), and e^{n}(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 surjectionI_{i,n}→I_{i,1} induces an isomorphism (I_{i,n})_{+}→(I_{i,1})_{+},
where (Ii,n)+denotes the wild inertia subgroup ofIi,n, i.e., (Ii,n)+is the trivial (resp. uniqueq-Sylow)
subgroup of I_{i,n} if the characteristic of k is 0 (resp. q). Now, for integers j ≥ 0 and n ≥ 0, define
a subgroup ( ˜I_{i,n})_{j} of I_{i,n} as follows: ( ˜I_{i,n})_{0} := I_{i,n} and, for j > 0, ( ˜I_{i,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 existsj_{0}≥0 such that ( ˜I_{i,n})_{j} ={1} forj > j_{0}.) Forj >0, thejth ramification
group (Ii,n)j of Ii,nmaps surjectively onto (Ii,1)dj/e_{i,n}(1)eby the above isomorphism (Ii,n)+→(Ii,1)+,
where ei,n(1) :=|I_{i,n}(1)| ([Se68, Chap. IV, Lem. 5]). From this, one obtains (Ii,n)j = ( ˜Ii,n)_{dj/e}_{i,n}_{(1)e}
forj≥0. (Note that this is clear forj = 0.)

Moreover, define ( ˜I_{i})_{j} to be the projective limit of {( ˜I_{i,n})_{j}}_{n≥0}. Thus, ( ˜I_{i})_{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