Towards a $ P $-Adic Uniformization of Abelian Varieties with Good Reduction

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logA 0 Ator(K) A(K) Lie(A)(K) 0

logA 0 Ator(Cp) A(Cp) Lie(A)(Cp) 0.

The group of points A(K) has a natural topology (see Subsection 2.14) whose completion is A(Cp) and which induces the discrete topology on Ator(K) and the natural, p-adic topology on Lie(A)(K). Let Ator(K) denote the torsion subgroup of A(K). We have the following decomposition

Ator(K)= Ap−tor(K) ⊕ Ap ′−tor(K) into the p-power torsion Ap−tor(K) and the prime to p-torsion Ap ′−tor(K). Fontaine has constructed a section s: A(Cp) Ap ′−tor(K) of the natural inclusion Ap ′−tor(K) ⊂ A(Cp) and we denote the (p) kernel of s by A (Cp) (see Section 6 or [Fon03] for more details on the construction of s). As such, we have the decompositions→ (p) A(Cp)= A (Cp) ⊕ Ap ′−tor(K) and (p) A(K)= A (K) ⊕ Ap ′−tor(K) (p) (p) where we let A (K) := A (Cp) ∩ A(K). (p) (p) Fontaine states, as a remarkable fact, that we can recover A (K), and so also A (Cp), as topological abelian groups with GK-action, from the knowledge of Ap−tor(K). Moreover, this implies that we can recover A(K), respectively A(Cp) from the knowledge of Ator(K) (cf. [Fon03, Proposi- tion 1.1]). 1.1. Main contributions. The present article has as objective to show that, under certain circum- stances, by changing the topology of A(p)(K), this new topological group can be determined in a different way from Ap−tor(K). First, we recall that in [Fon82], Fontaine constructed an integration map ϕA : Tp(A) Lie(A)(K) ⊗K Cp(1), and when this map was tensored with Cp, it realizes the Hodge–Tate comparison morphism. In particular, the map →

ϕA ⊗ 1Cp : Tp(A) ⊗Zp Cp Lie(A)(K) ⊗K Cp(1), is surjective and it has a large kernel, and to the best of our knowledge, the Fontaine integral has always been seen as a Hodge–Tate comparison morphism.→ A starting point for this article is the observation that if we do not tensor Tp(A) with Cp, then the Fontaine integral is often injective. To state the results precisely, we need to establish some notation. Let now K denote the maximal unramified extension of Qp, let A denote an abelian variety defined over some subfield F ⊂ K such that [F : Qp] < , with good reduction over F. Let A denote the Néron model of A over Spec(OF). G We present a proof, whose sketch was supplied by Pierre Colmez, that if Tp(A) K = 0, then Fontaine’s integration∞ map ϕA : Tp(A) Lie(A)(F) ⊗F Cp(1) is injective. There is another proof of this fact in the Appendix by Yeuk Hay Joshua Lam and Alexander Petrov (independently). In fact, we have more; to describe this,→ we need to briefly recall some definitions. We define the universal covering space of A(K)= A(OK) to be [p] [p] [p] B := A(O ) − A(O ) − · · · − A(O ) · · · A lim− K K K , 2 ← ← ← ← and we call an element u = (un)n>0 of BA a path. It is clear that BA is a GK-module which sits in the following exact sequence: ∼ α 0 Tp(A) = Tp(A) BA A(OK) 0, with α ((un)n>0) := u0 for all u = (un)n>0 ∈ BA. In Definition 2.12, we extend the classical Fontaine integral to a non-zero,→GF-equivariant map→ → →

ϕA : BA Lie(A)(OF) ⊗F Cp(1), and with this definition, it is clear that if a path u = (un)n>0 is periodic (i.e., there exists some k > 1 such that u0 = uk), then ϕA(u)(ω)=→ 0 (see Remark 2.13 for details). Our first result proves that the kernel of this extended Fontaine integral is precisely the periodic paths. Theorem A. Let A be an abelian variety over F with good reduction, and let A denote its Néron model. G Suppose that Tp(A) K = 0. Then, the kernel of the Fontaine integral ϕA extended to BA is precisely the subgroup of periodic paths of BA (Definition 3.6). G We give two applications of Theorem A. First, we show that if A satisfies Tp(A) K = 0, then (p) A (K), endowed with a new topology, can be determined in a different way from Ap−tor(K). ur Theorem B. Let K := Qp denote the maximal unramified extension of Qp. Let A denote an abelian variety defined over some subfield F ⊂ K such that [F : Qp] < , with good reduction over F. Let A denote the Néron model of A over Spec(OF). G Suppose that A satisfies Tp(A) K = 0. Then, there∞ exists a canonical, injective, continuous, GF- equivariant map (p) ,((ιA : A (K) ֒ (Lie(A)(F) ⊗F Cp(1)) /ϕA(Tp(A where the topology on A(p)(K) which makes this embedding continuous is the w-topology (Definition 6.1).

Moreover, an element x ∈ (Lie(A)(O→F) ⊗OF Cp(1)) /ϕA(Tp(A)) lies in the image of ιA if and only if x is crystalline (Deﬁnition 6.4). In particular, one can recover A(p)(K) from the triple

. ((Tp(A),Lie(A)(F) ⊗F Cp(1), ϕA : Tp(A) ֒ Lie(A)(F) ⊗F Cp(1) Remark 1.2. The fact that the field F of definition of the abelian variety A in the above theorem, is unramified is only used in Section 4.7 and in the Appendix.→ It seems clear that with more work all the results of this paper, suitably adjusted, should hold when F is a finite extension of Qp. Remark 1.3. In fact, strangely enough, we are able to prove analogues of the above results, on the kernel of the Fontaine integral and the uniformization, for the rigid analytic multiplicative group, that is G := Spm FhX, X−1i , with the multiplicative group law. Then the K-points of this group are G(K) = (O×, ·), i.e. the multiplicative group of O . K K This is strange as we do not have such a result for the multiplicative group over C. After some meditation on these results, we dare speculate that they should be shadows of the fact that such results would hold for Tate-curves, and what we see is the "crystalline" part of the logarithmic uniformization of some Tate-curve. Our second application concerns interesting properties regarding the ramification of the p- power torsion points of an abelian variety with good reduction over F. In particular, we prove the following result. Theorem C. Let A be an abelian variety over F with good reduction, and let A denote the formal group associated to the Néron model of A. Then there is n0 > 1 such that for every m > n0 and 0 6= P ∈ m (1) (1) A[p ](O), we have P ∈/ A(O ) where O is the kernel of the canonical derivativeb map on OF. Finally, we give a criterion for when the integer n from in Theorem C is equal to 1 (see Section b b 0 8.4, Definition 8.5, Theorem 8.8, and Proposition 8.9). 3 1.4. Related results. Theorem A shows that the zeros of the Fontaine integral coincide with the G periodic paths on BA when Tp(A) K = 0. The consideration of zeros of p-adic integrals (in the sense of Berkovich [Ber06], Coleman [Col85b], Coleman–de Shalit [CDS88], and Zarhin [Zar96]) has led to several deep results in Diophantine geometry. We refer the reader to [Cha41, Col85a, Col85b, Col87, Sto18, KRZB16, Kim05, Kim09] for details and the survey [KRZB18]. G Our Theorem B states that if A satisfies Tp(A) K = 0, then A(K) has a type of p-adic uniformization, which resembles the classical complex uniformization. The history of p-adic uniformization of abelian varieties is rich and beautiful, and we briefly exposit it below. The first work in this area was due to Tate [Tat71] who showed that an elliptic curve with multiplicative reduction is uniformized, as a rigid analytic space, by the rigid analytification of the multiplicative group. Later, Bosch–Lütkebohmert [BL85, BL84] constructed a topological universal cover for the rigid analytic space associated to an abelian variety and isolated a class of abelian varieties, namely those with toric reduction, whose topological uniformization resembles complex uniformization. Subsequent developments in p-adic geometry by Berkovich [Ber90, Ber93] gave rise to p-adic analytic spaces which have topological properties similar to those of complex man- ifolds. Using this theory, Berkovich [Ber99] showed that a smooth, connected Berkovich analytic space has a topological universal cover, and this result was later generalized by Hrushovshi– Loeser [HL16] to quasi-projective Berkovich analytic spaces. While Berkovich spaces have nice topological properties, adic spaces (in the sense of Huber [Hub94]) do not (e.g., they do not pos- sess paths). That being said, one can use the theory of perfectoid spaces [Sch12] and diamonds [SW20, Sch21] to construct certain pro-étale uniformizations of the adic space associated to an abelian variety (c.f. [BGH+20, Heu21]). To the best of our knowledge, Theorem B is the first result which gives a p-adic uniformization of abelian varieties with good reduction which is similar to the classical complex uniformization.

1.5. Future work. The firstquestiontoask is if our resultsextendto abelian varieties with semistable reduction over K, or even any abelian variety over K. The first author together with Nicola Maz- zari and Khai Hoan Nguyen have started to work on this and hope to soon report on it. In a different direction, Fontaine, in [Fon03], associates two interesting objects to Ap−tor(K): the (p) (p) topological group A (Cp), endowed with its GK-action and the rigid analytic group A , whose (p) Cp-points form the topological group A (Cp) above. Additionally, if the abelian variety has good reduction, we can associate to Ap−tor(K) a third object, namely a sheaf of abelian groups for the flat topology. The main goal of the article [Fon03] is to generalize loc. cit. Proposition 1 and these three constructions to motives. We believe that the constructions from Theorem B could also generalize to certain motives attached to algebraic varieties over p-adic fields.

1.6. Organization of paper. In Section 2, we recall the construction of Fontaine integration and discuss the role this integration theory plays in the context of Hodge–Tate and de Rham comparison isomorphisms for abelian varieties. In Section 3, we state Theorem A as well as how it relates to previous literature and to p-adic dynamical systems. In Section 4, we show that Theorem A can be reduced to the injectivity of the Fontaine integral restricted to the Tate module of the formal group and prove this, following a sketch provided by Pierre Colmez. The last three sections describe our applications of Theorem A. InSection6, weproveTheorem B, which is our p-adic uniformization result for A(p)(K), and in Section 7, we discuss the analogues of our works to the rigid multiplicative group. In Section 8, we give a different perspective on Fontaine integration, which allows us to prove Theorem C, and then in Section 8.4, we give a criterion to make Theorem C effective.

1.7. Conventions. We establish the following notations and conventions throughout the paper. 4 ur Fields. Fix a rational prime p > 2. Let K := Qp denote the maximal unramified extension of Qp, let K be a fixed algebraic closure of K, and let Cp denote the completion of K with respect to the unique extension v of the p-adic valuation on Qp (normalized such that v(p) = 1). For a tower of field extensions Qp ⊂ F ⊂ K, we denote by GK and respectively GF the absolute Galois groups of K and F respectively. We denote O := OK. We remark that working over K, as we do in this article, ur is not essential. One could start by fixing a finite extension L of Qp, and end-up working on L .

Abelian varieties. We will consider an abelian variety A defined over some subfield F ⊂ K such that [F : Qp] < , with good reduction over F. Let A denote the Néron model of A over Spec(OF) and also denote by A the formal completion of A along the identity of its special fiber, i.e. the formal group of A∞. We note that the formation of Néron models commutes with unramified base change. We will denote theb Tate module of A (resp. the Néron model A of A) by Tp(A) (resp. Tp(A)). We ∼ note that Tp(A) = Tp(A) as GF-modules. We recall that A is a formal group of dimension dim(A) and of height h which satisfies dim(A) 6 h 6 2 dim(A). b Acknowledgments. This article owes much to many people. We thank Robert Benedetto, Olivier brinon, Henri Darmon, Eyal Goren and Ralph Greenberg for helpful conversations and email ex- changes on topics related to this research. We are grateful to Pierre Colmez for sending us a sketch of the proof of Theorem4.8 and to Yeuk Hay Joshua Lam and Alexander Petrov for providing us with the proof presented in the Appendix. We also thank Pierre Colmez and Jan Nekovar for pointing out some errors in earlier drafts of this paper.

2. Fontaine integration for abelian varieties with good reduction In this section, we recall the construction of the Fontaine integration and the extension of this integration theory to a certain universal cover of an abelian variety. We also describe several topological aspects of the integration theory and discuss how these integration theories realize the Hodge–Tate and de Rham comparison isomorphisms for abelian varieties.

2.1. The differentials of the algebraic integers. First, we recall for the reader’s convenience the ur notation established above. Let K := Qp denote the maximal unramified extension of Qp, let K be a fixed algebraic closure of K, and let Cp denote the completion of K. Let GK denote the absolute Galois group of K. We denote O := OK. Fix a finite extension F of Qp in K. In [Fon82], Fontaine studied a fundamental object related to these choices, namely the O- module Ω := Ω1 =∼ Ω1 of Kähler differentials of O over O , or over O . The O-module Ω O/OK O/OF K F is a torsion and p-divisible O-module, with a semi-linear action of GF. Let d: O Ω denote the canonical derivation, which is surjective. Important examples of algebraic differentials arise as follows: Let (εn) denote→ a compatible sequence of primitive pth roots of unity in K. Then

dεn dε + dεn = d(log ε ) ∈ Ω and p n 1 = . ε n ε ε n n+I n Next, we recall a theorem of Fontaine.

Theorem 2.2 ([Fon82, Théorème 1’]). Let (εn) denote a compatible sequence of primitive pth roots of unity in K. The morphism ξ: K(1) Ω deﬁned by

dεr → ξ(α ⊗ (εn)n)= a εr 5 r where α = a/p for some a ∈ O is surjective and GK-equivariant with kernel 1 ker(ξ)= a := x ∈ K : v(x) > − . K p − 1

∼ ∼ ∼ Moreover, Ω = K(1)/aK(1) = (K/aK)(1) and Vp(Ω)= HomZp (Qp, Ω) = Cp(1). Theorem 2.2 implies the following:

p p p T (Ω) ⊗ Q := lim Ω[pn] ⊗ Q =∼ lim Ω Ω · · · Ω · · · ⊗ Q =∼ C (1) p Zp p − Zp p − Zp p p n ! as GF-modules. ← ← ← (1) ← ← We denote by O := ker(d), the kernel of d, which is an OK-sub-algebra of O. Indeed, if a, b ∈ O(1), then d(ab) = ad(b)+ bd(a) = 0, and so ab ∈ O(1). In order to better understand O(1), we recall a construction from [IZ99]. Definition 2.3. Let a ∈ O. Let L/K be a finite extension which contains a, let π be a uniformizer of L, and let f ∈ OK[x] be such that a = f(π). Then, define f′(π) δ(a) := min v , 0 ∆ L/K where ∆L/K denotes the different ideal of L/K. Note that δ does not depend on π, f, or F, and so it defines a function δ: O (− , 0]. Lemma 2.4 (Properties of δ). The function δ from Definition 2.3 satisfies the following properties. → ∞ (1) If a, b ∈ O, then δ(a + b) > min(δ(a), δ(b)), and if δ(a) 6= δ(b), then we have equality. (2) If a, b ∈ O, then δ(ab) > min(δ(a)+ v(b), δ(b)+ v(a)). ′ (3) If f ∈ OK[x] and α ∈ O, then δ(f(α)) = min(v(f (θ)) + δ(θ), 0). (4) If x, y ∈, then xdy = 0 if and only if v(x)+ δ(y) > 0. (5) For a ∈ O, δ(a)= 0 if and only if a ∈ O(1). (6) The formula δ(adb) := min(v(a)+ δ(b), 0) is well-defined and give a map δ: Ω (− , 0], which makes the obvious diagram commutative. We will use the follow properties of δ in our study of the Fontaine integral. → ∞

Lemma 2.5 ([IZ99, Lemma 2.2]). Let a, b ∈ O be such that δ(a) 6 δ(b). Then there exists c ∈ OK[a,b] such that cda = db. Proposition 2.6 ([IZ99, Theorem 2.2]). Let L/K be an algebraic extension. Then L is deeply ramified (loc. cit. Definition 1.1) if and only if δ(OL) is unbounded. To conclude our discussion on differentials of the algebraic integers, we note that the derivation map d: O Ω is not continuous with respect to the p-adic topology on O and the discrete topology on Ω. Nevertheless, there is another topology on O for which d is continuous, namely the topology→ defined by the valuation w: O − Z ∪ { } defined by w(a) := sup{n ∈ Z | a ∈ pnO(1)}, where O(1) = ker(d). We summarize this result in the following lemma. Lemma 2.7. Consider O with the topology defined→ by the∞ valuation w and endow Ω with the discrete topology. Then, the derivation map d: O Ω is continuous. 2.8. A universal cover of an abelian variety. Classically, the Fontaine integral is defined as a map → ϕA : Tp(A) Lie(A)(F) ⊗F Cp(1). For our purposes, we will want to integrate along a larger collection of paths, which leads to the following definition. We note that this construction→ appears in [CI99, Part II, Section 3]. 6 Definition 2.9. We define the universal covering space of A(K) to be

[p] [p] [p] [p] [p] [p] B := A(O) − A(O) − · · · − A(O) · · · = A(K) − A(K) − · · · − A(K) · · · A lim− lim− where the equality← comes from← the properness← of A. We call an element← u =← (un)n>0←of BA a path. ← ← We immediately have that BA is a GF-module which sits in the following exact sequence: α 0 Tp(A)= Tp(A) BA A(O) 0, (2.1) with α (u) := u0 for all u = (un)n>0 ∈ BA. → → → → Remark 2.10. The universal covering space from Definition 2.9 is similar to the construction of the perfectoid universal cover of an abelian variety (see e.g., [PS16, Lemma 4.11] and [BGH+20]). We refrain from using this terminology as we work over the field K, which is not perfectoid, and hence BA cannot come from any perfectoid space. We also mention that [Col98, Appendice B] discusses universal covering spaces of abelian varieties over p-adic fields. 2.11. The definition of Fontaine integration. We are now ready to define Fontaine integration. Let H0(A, Ω1 ) and respectively Lie(A)(O ) denote the O -modules of invariant differentials A/OF F F on A and respectively its Lie algebra.

0 1 Deﬁnition 2.12. Let u = (u ) N ∈ B and ω ∈ H (A, Ω ). Each u ∈A(O) corresponds to n n∈ A A/OF n a morphism un : Spec(O) A, and hence we can pullback ω along this map giving us a Kähler ∗ ∗ differential un(ω) ∈ Ω. The sequence (un(ω))n>0 is a sequence of differentials in Ω satisfying ∗ ∗ ∼ pun+1(ω)= un(ω), and hence→ deﬁnes an element in Vp(Ω) = Cp(1). The Fontaine integration map

ϕA : BA Lie(A)(OF) ⊗OF Cp(1) is a non-zero, GF-equivariant map defined by → ∗ ∼ ϕA(u)(ω):=(un(ω))n>0 ∈ Vp(Ω) = Cp(1). Remark 2.13. Using Theorem 2.2 and the function δ from Definition 2.3, we can give an alternative 0 1 description of the Fontaine integration map. Let u = (u ) > ∈ B and ω ∈ H (A, Ω ). Each n n 0 A A/OF un ∈ A(O) corresponds to a morphism un : Spec(O) A, and hence we can pullback ω along ∗ this map giving us a Kähler differential un(ω) ∈ Ω. > > ∗ For every n 0, there is a maximal m(n) 0 such that→ un(ω)= αn(dεm(n)/εm(n)) with αn ∈ O m(n) where εm(n) is some primitive p -th root of unity. To see this, we first note that dε 1 δ r =−r − ε pr(p − 1) r for any primitive pr-th root of unity. This result follow from the definition of δ and a result of Tate [Tat67, Proposition 5] on the valuation of the different ideal of K(εr)/K. By taking m(n) = ∗ −[δ(un(ω))] where [x] denotes the greatest integer of the real number x, we can use Lemma 2.4.(6) and Lemma 2.5 to deduce the above equality. Now using Theorem 2.2, we have that n−m(n) ϕA(u)(ω)= lim p αn ∈ Cp. n→∞ G Moreover, using the definition of δ and this above interpretation, we can see that if u ∈ (BA) K (i.e., if u is an unramified path), then ϕA(u)(ω)= 0. Indeed, it is clear from the definition of δ that m(n)= 0. 7 2.14. Topological aspects of the extended Fontaine integral. Recall the notation from Subsection 1.7. We have two GF-modules, namely A(K)= A(O) and BA, which was defined in Definition 2.9. A priori, A(K) = A(O) and BA are just abelian groups, but we can endow them with natural topologies to enhance them to topological abelian groups as follows. Fontaine defines the following natural topology on A(K), namely the coarsest topology for which the maps A(O) A(O/pnO) are continuous, for all n > 1, with the discrete topology on the target. We denote A(O) with this topology by AFo(O), which makes A(K) = A(O) into a topological abelian group→ and induces the discrete topology on Ator(K), the subgroup of torsion points of A(K) and the p-adic topology on the points of A. Let ψA : A(O) Lie(A)(OF) ⊗OF Ω ∗ denote the Fontaine integral i.e., defined by sending a ∈A(O) to ψA(a)(ω) := a (ω) ∈ Ω where ω ∈ H0(A, Ω1 ). As the derivative map d is not continuousb with respect to the natural p-adic A/OF → Fo topology on O, we see that the map ψA is not continuous for A (O) and the discrete topology on Lie(A)(OF) ⊗OF Ω. In order to get continuity of the Fontaine integral ψA and the extended Fontaine integral ϕA, we will need to define a new topology on which resembles the w-topology from Lemma 2.7. Definition 2.15. The w-topology on A(O) = A(K) is defined to be the coarsest linear topology for which A(O(1)) ⊂ A(O) is open and all the maps A(O(1)) A(O(1)/pnO(1)) are continuous with respect to the discrete topology on the target. In other words, this will be the linear n (1) ∼ topology on A(K) such that a base of open neighborhoods of 0 ∈→A(K) is given by A(p O ) = n (1) g ∼ n (1) g > p O , ⊕A = p O , ⊕A , for all n 1 where we denote by ⊕A the formal group ( ) ( ) b law of Aand by ⊕Athe group law in A K . Moreover, if x ∈ A O is a point, then the subsets n (1b) b w x ⊕A A p O for n > 1 of A(O) give a base of open neighborhoods of x in A (O). We denotedb A(O) with the w-topology by Aw(O). b Remark 2.16. The w-topology on A(O) induces the w-topology on A(O(1)), and later, we will see GK that when Tp(A) = 0, it induces the discrete topology on Ator(K). b w To conclude this discussion, we will show that the action of GF on A (O) and the Fontaine w integral ψA defined on A (O) are continuous. w Lemma 2.17. The action of GF on A (O) is continuous. Fo Proof. We know that the action of GF on A (O) is continuous. Moreover, the topologies induced Fo w on A(OL)= A(L), for a finite extension L/F in K, by A (O) and by A (O) are the same, therefore the conclusion follows. Lemma 2.18. The Fontaine integral ψ : Aw(O) × H0(A, Ω1 ) Ω defined by (x, ω) x∗(ω) ∈ A A/OF Ω is continuous. Proof. We recall that on Aw(O) we have the w-topology, on H0→A, Ω1 we have→ the p-adic A/OF topology, and on Ω we have the discrete topology. Let x ∈ A(O) and ω ∈ H0 A, Ω1 and denote by η := x∗(ω) ∈ Ω. Let U := ψ−1(η) ⊂ A/OF A w 0 1 A (O) × H A, Ω . We claim U is a neighborhood of (x, ω). For this let Vx := x ⊕A A/OF m (1) A p O for some m > 1 be a neighbourhood of x. Let ω1, . . . , ωg denote an OF-basis of 0 1 n ∗ H A, Ω and let n > 1 be an integer such that p x (ωi) = 0 for i = 1, 2, . . . , g. Then for A/OF 0 1 n ∗ n 0 1 anyb β ∈ H A, Ω , we have p x (β)= 0. Let Vω := ω + p H A, Ω denote the neigh- A/OF A/OF borhood of ω. We claim that Vx × Vω ⊂ U. m (1) n Let (y, γ) ∈ Vx × Vω, i.e. y = x ⊕A z, with z ∈ A p O and γ = ω + p β, with β ∈ H0 A, Ω1 . We have A/OF ∗ n ∗ n ∗ n b ∗ n ∗ ∗ (x ⊕A z) (ω + p β)= x (ω + p β)+ z (ω + p β)= x (ω)+ p x (β)= x (ω)= η, 8 which gives our desired result. Note that the first equality holds because the differential ω + pnβ is invariant. For the second equality, we have, denoting δ := ω + pnβ ∈ H0 A, Ω1 , that A/OF g g ∗ ∗ ∗ z (δ)= z (δ|A) = (z1, z2,..., zg) Fi(X1,..., Xg)dXi = Fi(z1, . . . , zg)dzi = 0 i=1 ! i=1 b X X where A = Spf(OF[[X1, . . . , Xg]]) and Fi(X1, . . . , Xg) ∈ OF[[X1, . . . , Xg]]. The final equality holds because z ∈ A pmO(1) . b Definition 2.19. We define the w-topology on B to be the projective limit topology with the w- b A topology defined on A(O). Using Lemma 2.17 and Lemma 2.18, it is a simple exercise, which we leave to the reader, to show that ϕA : BA Lie(A)(OF) ⊗OF Cp(1) is GF-equivariant and continuous where BA has the w-topology and Lie(A)(OF) ⊗OF Cp(1) is endowed with the p-adic topology. → 2.20. Comparison isomorphisms via p-adic integration. To conclude this section, we discuss the role of Fontaine integration in the Hodge–Tate and de Rham comparison isomorphisms for abelian varieties. Fontaine [Fon82] originally defined this integration map in order to re-prove the Hodge–Tate comparison isomorphism for abelian varieties. More precisely, let A∨ denote the dual abelian variety, which is also defined over F and with good reduction and we denote by A∨ its Néron model over OF. See[BLR90, Theorem 8.4.5] for details. We write

ϕA ⊗ 1Cp : Tp(A) ⊗Zp Cp Lie(A)(F) ⊗F Cp(1) ∨ ∨ ∨ ϕA ⊗ 1Cp : Tp(A ) ⊗Zp Cp Lie(A )(F) ⊗F Cp(1), → which are both surjective maps. By duality, the Weil pairing, and further arguments, we obtain the following isomorphism →

∨ ∼ ∨ Tp(A) ⊗Zp Cp = Lie(A )(F) ⊗F Cp ⊕ (Lie(A)(F) ⊗F Cp(1)) , which is the Hodge–Tate decomposition of Tp(A). We also mention work of Coleman [Col84], which complements [Fon82] and gives another proof of the Hodge–Tate decomposition. In addition to the Hodge–Tate decomposition, there is a de Rham comparison isomorphism for 1 an abelian variety A. More precisely, let HdR(A) denotes the first de Rham cohomology group of the abelian variety A; it has a natural filtration expressed by the exact sequence: 0 1 1 0 H (A, ΩA/F) HdR(A) Lie(A)(F) 0. In [Col92], Colmez defined a p-adic integration pairing, now called Colmez integration, which is a functorial, perfect pairing→ → → → 1 + 2 h·, ·iCz : Tp(A) × HdR(A) BdR/I , which realizes the de Rham comparision isomorphism. Here I ⊂ B+ is the maximal ideal of B+ → dR dR and the above pairing is GF-equivariant in the first argument and respects filtrations in the second 0 1 1 argument. By restricting to H (A, ΩA/F) ⊂ HdR(A), Colmez integration induces a pairing: 0 1 2 ∼ h·, ·iF : Tp(A) × H (A, ΩA/F) I/I = Cp(1), which gives a map Tp(A) Lie(A)(F) ⊗F Cp(1). By[Col92, Proposition 6.1], this map coincides → with Fontaine’s integration (ϕA)|Tp(A), when A has good reduction. → 9 3. The result on the zeroes of the Fontaine integral

In this section, we investigate the kernel of ϕA : BA Lie(A)(F) ⊗F Cp(1). We recall that the abelian variety is defined over F, a finite extension of Qp contained in K and denote IF := GK, the inertia subgroup of GF. As stated in Remark 2.13,→ the Fontaine integral will vanish on an G element u = (un)n>0 ∈ BA if un ∈ OK for all n i.e., if the path u lies in (BA) K . It is natural to speculate whether the Fontaine integral restricted to Tp(A) will be injective away from unramified paths which live in Tp(A). To make this question easier to study, we will impose the following assumption throughout. G Assumption 3.1. We assume that the abelian variety A defined over F satisfies Tp(A) K = 0. Remark 3.2. Assumption 3.1 is not very restrictive. For example, if A is an elliptic curve, Assumption 3.1 is equivalent to the property that A does not have CM by a quadratic imaginary field M, in which p is split (see e.g., [Oze10, Theorem 2.11] and [Ser98, A.2.4] for details). We refer the reader to [Ser98, A.2.3] for further discussion of the relationship between complex multiplication and Assumption 3.1.

We begin our analysis of ker(ϕA) by identifying two of its subgroups. First, we again remark G that (BA) K ⊂ ker(ϕA). Indeed, this follows from the above discussion, or one may deduce it from the fact that GK (Lie(A)(OF) ⊗OF Cp(1)) = 0, which is a consequence of a result of Tate [Tat67, Theorem 2]. We can easily determine another signiﬁcant subgroup of ker(ϕA).

Deﬁnition 3.3. A path u = (un)n>0 ∈ BA is periodic if there exists some k > 1 such that un = un+k for all n > 0. More precisely, a periodic path is of the form

k−1 k−2 k−1 k−1 u := u0, [p ]u0, [p ]u0 . . . , [p]u0, u0, [p ]u0, . . . , [p]u0, u0, [p ]u0 . . . , (3.1)

k where, let us remark that [p ]u0 = u0.

Lemma 3.4. Let u = (un)n>0 ∈ BA be a periodic path. Then u0 is a prime-to-p torsion point of A(O) G and u ∈ (BA) K . k k Proof. Let k > 1 be the smallest integer such that uk = u0. Then u0 = [p ]uk = [p ]u0 which k k implies [p − 1]u0 = 0 and therefore u0 ∈A[p − 1](O). Since A has good reduction at p, one can use formal groups to deduce that u0 ∈A(OK). Now, the latter claim follows from the description of a periodic path in (3.1). Lemma 3.5. Suppose that A satisﬁes Assumption 3.1, and let u be a prime-to-p torsion point of A. Then, G there is a unique periodic path u = (un)n>0 ∈ (BA) K such that u0 = u. G G Proof. If v := (vn)n>0 ∈ (BA) K is a path such that v0 = u0, then u − v ∈ Tp(A) K = 0 by (2.1) and Assumption 3.1.

GK Deﬁnition 3.6. We deﬁne per(BA) ⊂ (BA) to be the subgroup of periodic sequences of BA.

Lemma 3.7. The subgroup per(BA) is isomorphic to the subgroup of prime-to-p torsion points on BA which itself is isomorphic to the subgroup of prime-to-p torsion points on A(O).

Proof. We first note that the prime-to-p torsion on BA is isomorphic to the prime-to-p torsion on A(O). Indeed, this follows because the multiplication-by-p map is an isomorphism on A[n](O) for every n coprime to p. The result now follows from Lemma 3.4. With these definitions established, we can now state our main result. 10 Theorem 3.8 (=Theorem A). Let A be an abelian variety over F, with good reduction, and let A denote its Néron model. Suppose that A satisfies Assumption 3.1. Then we have that

GK per(BA) = (BA) = ker(ϕA). Below, we make two remarks relating Theorem 3.8 to previous literature on determining the zeros of p-adic integrals and to p-adic dynamical systems. Remark 3.9. We wish to highlight the similarities between our Theorem 3.8 and [Col85b, Theorem 2.11]. The result of Coleman says that the torsion points on an abelian variety correspond to the set of common zeros of p-adic abelian integrals of the first kind on the abelian variety. Our Theorem 3.8 shows that for an abelian variety A over K satisfying Assumption 3.1, the common zeros of the Fontaine integral are precisely the torsion points on BA. Indeed, recall that Lemma 3.7 shows the subgroup per(BA) coincides with the prime-to-p torsion on BA, and since BA will have no p-power torsion by construction, the statement follows. Remark 3.10. We want to give an interpretation of Theorem 3.8 in terms of p-adic dynamical systems. First, we note that Theorem 3.8 reduces to showing that for u = (un)n>0 ∈ BA a non- periodic path, the valuation of αn from Remark 2.13 is bounded with respect to n for all n ≫ 0. Indeed, such a result will tell us that ϕA(u)(ω) is non-zero and hence not in the kernel. In order to study v(αn), we use the function δ and Lemma 2.4.(6). Recall from Remark 2.13 that we precisely r know the value of δ(dεr/εr) where εr is a primitive p -th root of unity, and so if we can compute ∗ (or more realistically, bound) δ(un(ω)), then we can understand v(αn). ∗ To determine δ(un(ω)), we need to understand various aspects of the field extensions K(un)/K for all n > 0. In particular, we would like to find explicit uniformizers for K(un)/K and to compute the different ideal of the extension K(un)/K. Both aspects are very difficult in general; we refer the reader to [Viv04, BL20] (resp. [Sve04, AHM05, Ber14, Ber16, Nor18]) for a discussion of these topics, respectively. That being said, understanding the ramification behavior of the extensions K(un)/K is a question of interest in the field of p-adic dynamical systems. Our Theorem 3.8 shows that the algebraic extension K∞ = n>0 K(un) is deeply ramified, or equivalently by Proposition 2.6, the values δ(O ∞ ) are unbounded. In fact, it says more. Using K S Remark 2.13, we can see that Theorem 3.8 states that the function |δ(OK(un))| grows like n plus a term which is bounded with respect to n for all n ≫ 0, and unwinding the definition of δ, this says the valuation of the different ideal of K(un)/K is at least n. If A is an elliptic curve and u = (un) ∈ Tp(A), then this type of result can be deduced using formal Weierstrass preparation, workof Katz[Kat73, Section 3.6], and some basics of p-adic dynamics (see e.g., [Ben19]). However, when A has dimensionb > 1, proving such a result using dynamical methods seems beyond reach due to the lack of techniques to analyze p-adic multivariable power series.

4. The proof of Theorem 3.8 In this section, we first describe the reduction of the proof of Theorem 3.8 to a statement about injectivity of the Fontaine integral restricted to the Tate module of the formal group associated to our abelian variety and then prove this injectivity following an idea of P. Colmez. 4.1. Reduction of Theorem 3.8 to the setting of the Tate module of the formal group. Let A/F be an abelian variety with good reduction, A its Néron model over OF and we assume that A satisfies Assumption 3.1. Let A denote the formal completion of A along the identity of its special fiber, i.e. the formal group of A. Then, if we denote by g the dimension of A over Spec(F), we have that b ∼ A = Spf (OF[[X1, . . . , Xg]]) as formal schemes, with a formal group-law given by g power series b A1(X1, . . . , Xg, Y1, . . . , Yg), . . . , Ag(X1, . . . , Xg, Y1, . . . , Yg). 11 b b Let Tp(A) ⊂ Tp(A) be the Tate module of the formal group and let us recall that the restriction to the formal group defines an isomorphism H0(A, Ω1 ) =∼ Inv(A), where the second O -module A/OF F is the moduleb of invariant differentials of the formal group. We also have natural isomorphisms ∼ Lie(A)(OF) = Lie(A)(OF), where the second module is the Lie algebrab of the formal group, i.e. its tangent space at the origin. Let (ϕ ) : Tpb(A) Lie(A)(O ) ⊗ Cp(1) denote the restriction of ϕ to Tp(A). The main A |Tp(A) F OF A result of this subsection proves that if (ϕ ) is injective, then Theorem 3.8 holds. A |Tp(A) b b → b b Theorem 4.2. Assume that A satisfies Assumptionb 3.1 and is such that ker((ϕ ) ) = 0. Then we A |Tp(A) have b (1) The restriction (ϕA)|Tp(A) : Tp(A) Lie(A)(OF) ⊗OF Cp(1) is injective. GK (2) We have ker(ϕA)= per(BA) = (BA) i.e., Theorem 3.8 holds. → Proof of Theorem 4.2.(1). We first prove (ϕA)|Tp(A) is injective. If we denote by A(p) and respectively A(p), the p-divisible groups attached to the abelian scheme A and respectively its formal group scheme, then we have an exact sequence of p-divisible groups b 0 A(p) A(p) A(p)/A(p) 0.

As A(p) is identiﬁed with the connected sub-p-divisible group of A(p), the p-divisible group → b → → b → A(p)/A(p) is an étale p-divisible group over OK, therefore it is isomorphic, as p-divisible groups, 2g−h to (Qbp/Zp) , where h is the height of A. The exact sequence of p-divisible groups above deﬁnesb and exact sequence of GK-representations b ψ 2g−h 0 Tp(A) Tp(A) (Zp) 0, (4.1)

Let e1, e2, . . . , e2g denote a Zp-basis of Tbp(A), such that e1, . . . , eh is a basis of Tp(A) and the images 2g−h → → → → ψ(eh+1), . . . , ψ(e2g) in (Zp) give a Zp-basis of this module. We can describe the action of GK on the basis elements as follows: if σ ∈ GK and 0 6 i 6 2g − h , then b

σ(eh+i)= eh+i + ξi(σ) where ξi : GK Tp(A) is a continuous 1-cocycle, for all 0 6 i 6 2g − h.

Suppose for the sake of contradiction that (ϕA)|Tp(A) is not injective, and let b → 2g−h

0 6= x = f + aieh+i, = Xi 0 6 6 denote an element of ker((ϕA)|Tp(A)) with f ∈ Tp(A) and ai ∈ Zp for all 0 i 2g − h. Let σ ∈ GK and consider the relations: b ( ) ( ) + 2g−h ( ) ( ) = ϕA |Tp(A) f i=0 ai ϕA |Tp(A) eh+i 0 P ( ) ( ) + 2g−h ( ) ( ( )) + 2g−h ( ) ( ) = σ ϕA |Tp(A) f i=0 ai ϕA |Tp(A) ξi σ i=0 ai ϕA |Tp(A) eh+i 0. By substracting these rows,P one obtains P

2g−h

(ϕA)|Tp(A) f − σ(f)− aiξi(σ) = 0. i=0 ! 12 X As (ϕ ) is injective, it follows that A |Tp(A)

b 2g−h aiξi(σ)= f − σ(f), for all σ ∈ GK, = Xi 0 which implies that the cohomology class

2g−h

ai[ξi]= 0 = Xi 0 1 in H (GK, Tp(A)), where we denoted [ξi] the class of the cocycle ξi. The exact sequence (4.1) gives the long exact sequence in cohomology: b GK 2g−h δ 1 0 = Tp(A) (Zp) H (GK, Tp(A)) G where Tp(A) K = 0 by Assumption 3.1 and δ (ψ(eh+i)) = [ξi] for all 0 6 i 6 2g − h. Therefore → → b 2g−h 2g−h

δ(ψ(x)) = δ aiψ(eh+i) = ai[ξi]= 0. − ! = Xi 0 Xi 0

As δ is injective, it follows that ψ(x)= 0, i.e. x = f ∈ Tp(A), and since (ϕA)|Tp(A)(x) = (ϕA)|Tp(A)(f)= 0, we have x = 0. This is a contradiction, which proves our desired result. b Before we prove Theorem 4.2.(2), we need a lemma.

Lemma 4.3. Keep the notations and assumptions from Theorem 4.2. Let u ∈ BA be such that u0 = α(u) ∈ G A(OL), for L/F a ﬁnite extension in K. If ϕA(u)= 0, then u ∈ (BA) L , where GL = Gal(K/L).

Proof. First note that for every σ ∈ GL we have that α (σ(u)− u) = 0, i.e. σ(u)− u ∈ Tp(A).

Theorem 4.2.(1) implies that (ϕA)|Tp(A) is injective. Therefore if (ϕA)(u) = 0, then we have

(ϕA)|Tp(A)(u − σ(u)) = ϕA(u)− σ(ϕA(u)) = 0, and hence σ(u)− u = 0 for all σ ∈ GL. Thus, for all σ ∈ GL we have σ(u)= u.

Proof of Theorem 4.2.(2). We already know that per(BA) ⊂ ker(ϕA), so it sufﬁces to prove the re- verse inclusion. We break the proof up into three cases.

Case 1. Let u ∈ ker(ϕA) such that u0 := α(u) ∈ A(O)[m] for m > 1 with (m, p) = 1. Then G G u0 ∈ A(OK) and by Lemma 4.3, we see that u ∈ (BA) K . However, the only element u ∈ (BA) K with u0 ∈A(OK)[m] for m above is the periodic sequence associated to u0, and so u ∈ per(BA).

Case 2. Let us now supposethat u ∈ ker(ϕA) with u0 = α(u) ∈A(O) such that u0 is a torsion point a a a of A of order a power of p, say u0 ∈A(O)[p ] for some a > 1. Then p (u) = ([p ]un)n>0 ∈ Tp(A) a a a and ϕA(p (u)) = p ϕA(u) = 0. As (ϕA)|Tp(A) is injective, we have that p (u) = 0 in BA. Since BA[p]= 0, we see that u = 0.

Case 3. Finally, let us suppose that u ∈ ker(ϕA) with u0 = α(u) not a torsion point of A(O).

Observe that u ⊗ 1Qp ∈ BA ⊗Z Q whose image via α ⊗ 1Q in A(O) ⊗Z Q is u0 ⊗ 1 6= 0. We have a natural exact sequence of Qp-vector spaces, with continuous action by GK, namely

0 Vp(A) BA ⊗Z Q A(O) ⊗Z Q 0. (4.1) ∼ Notice that as Tp(A) is naturally a Zp-module, we have that Vp(A) := Tp(A) ⊗Z Q = Tp(A) ⊗Zp Qp. → → 13 → → Let us now suppose that u ∈ ker(ϕA) has the property that u0 = α(u) ∈A(OL) for some ﬁnite extension L of F, in K and that, as stated above, 0 6= u0 ⊗ 1 ∈ A(O) ⊗Z Q. We denote, as usual GL = Gal(K/L) and let αL : BA,L A(OL) be the ﬁber product of the diagram of GL-modules: α BA − A(O) → ∪ → A(OL) In other words we have a cartesian, commutative diagram of GL-modules, with exact rows α⊗1 0 Vp(A) BA ⊗Z Q A(O) ⊗Z Q 0 || ∪ ∪ αL⊗1 0 → Vp(A) → BA,L ⊗Z Q → A(OL) ⊗Z Q → 0 It is easy to see that we have, on the one hand → → → → GL (BA) ⊂ BA,L ⊂ BA,

GL GL therefore we have BA,L ⊗Z Q = BA ⊗Z Q . On the other hand the exact sequence

0 Vp(A) BA,L ⊗Z Q A(OL) ⊗Z Q 0 is an exact sequence of ﬁnite Qp-vector spaces with continuous GL-action, whose long exact GL- cohomology sequence reads→ as → → →

GL αL⊗1 ∂ 1 (BA ⊗Z Q) − (A(OL) ⊗Z Q) − H (GL, Vp(A)). By [BK07, Example 3.11], the map ∂ is injective, and therefore, we have that → → GL αL ⊗ 1 (BA ⊗Z Q) = 0. Thus, there is no GL-invariant element of BA ⊗ Q which has image under α ⊗ 1 equal to u0 ⊗ 1Q 6= 0, and so u cannot be GL-invariant. However, we note that this contradicts Lemma 4.3. Therefore, we can conclude and say that ker(ϕA)= per(BA), as desired. As very easy consequences of the above considerations, we have the following examples of classes of abelian varieties satisfying Theorem 3.8. We recall that Fontaine [Fon82] proved that the morphism

(ϕA)|Tp(A) ⊗ 1Cp : Tp(A) ⊗Zp Cp Lie(A)(OF) ⊗OF Cp(1) is surjective and it defines the Hodge–Tate decomposition of Tp(A). In particular, the morphism ϕA is non-zero. → Lemma 4.4. Let A be an elliptic curve over F with good reduction satisfying Assumption 3.1. Then, Theorem 3.8 is true for A. Proof. First, we consider the case where A has ordinary reduction, i.e. its formal group has dimension 1 and height 1. As ϕA is non-zero, it will be injective on Tp(A) which has dimension 1, and hence Theorem 4.2 implies that A satisfies Conjecture 3.8. If A has super-singular reduction, its formal group has dimension 1 and height 2, and we have Tp(A) =b Tp(A), which moreover is an irreducible GF-module. As ϕA is a non-zero, GF-equivariant map, it follows that it is injective on Tp(A), and so Theorem 4.2 again says that A satisfies Conjecture 3.8. b Lemma 4.5. Let A be an abelian variety over F that is isomorphic over F to a product of abelian varieties ∼b s A = i=1 Ai. Suppose that A and Ai for i = 1, . . . , s have good reduction over OF and the isomorphism extends to an isomorphism over OF of Néron models. Then, we have that A satisfies Assumption 3.1 and Q Theorem 3.8 if and only if every Ai satisfies Assumption 3.1 and Theorem 3.8, for all i = 1, . . . , s. 14 ∼ s ∼ s Proof. This follows immediately because Tp(A) = i=1 Tp(Ai) and Lie(A)(OF) = i=1 Lie(Ai)(OF), s and the integration maps admit a decomposition as ϕA = ϕA . Q i=1 i Q Lemma 4.6. Let A over F be an abelian variety with good ordinaryQ reduction which satisfies Assumption 3.1. Then, Theorem 3.8 is true for A.

Proof. The formal group A of the Néron model A is isomorphic, over Spf(OF), to a product of copies of the formal multiplicative group, which has dimension 1 and height 1. The restriction of ϕA to the Tate module ofb each component of the formal group is non-zero, and hence injective. Therefore, the restriction of ϕA to the Tate module of the formal group of A is injective and so Theorem 4.2 implies that A satisfies Theorem 3.8. 4.7. Injectivity of Fontaine integral restricted to Tate module of formal group. We now present a proof of the injectivity of Fontaine’s integration map, restricted to the Tate module of the formal group of the abelian variety. The outline of this proof was communicated to us by Pierre Colmez. We briefly recall the relevant notation. Let A be the formal group of an abelian variety A, with good reduction over a finite extension F of Qp, in K, where we recall that K is the maximal unramified extension of Qp inside some algebraicb closure Qp. In this subsection, we do not assume G Assumption 3.1, i.e. we do not assume that that Tp(A) K = 0. 1 Let D := HdR(A) denote the Dieudonné module of A over F, which is an admissible filtered, ∼ ∨ Frobenius module over F. Moreover, D = Dcris(V), for V := Vp(A) , by which we mean the Qp- dual of the GF-representationb Vp(A). Let φD denote theb linear Frobenius of D, i.e. if [F : Qp] = r abs r abs b then φD is the linear map (φD ) , where φD is the absolute Frobenius on D. Wedenoteby W ⊂ D the subspace which determines theb filtration of D, that is: Fili(D)= D for i 6 0, Fil1(D)= W and Filj(D)= 0 for j > 2. For every filtered, Frobenius submodule D′ of D, we define the Hodge and Newton numbers of ′ ′ ′ D , tH(D ), respectively tN(D ) in terms of the filtration and the dimensions of its graded quotients and respectively, in terms of the normalized slopes of the Frobenius φD ′ . We say that D is weakly ′ ′ ′ admissible if for every D ⊂ D sub-filtered, Frobenius module we have tN(D ) > tH(D ), with equality if D′ = D. In particular, we recall that as D is admissible, it is weakly admissible. Let us consider the integration pairing as in [Col92, Section 7] cris 1 ∨ + ϕA : Tp(A) − HdR(A) ⊗F Bcris, which induces on the one hand the crystalline integration on the Tate module of the formal group: → ϕcris : T (A) − D∨ ⊗ B+ A p F cris. and on the other hand Fontaine’s integration on the Tate module of the formal group b b → ∨ ϕA : Tp(A) − W ⊗F Cp(1). where we identify (ϕ ) with ϕ . The main result of this subsection is the following. A |Tp(A) A b b → ∨ b b Theorem 4.8. The map ϕA : Tp(A) − W ⊗F Cp(1) is injective. Proof. The outline of this proof was supplied by Pierre Colmez. b b → Let x ∈ Tp(A) be such that ϕA(x)= 0. This means that for every ω ∈ W, + b 2 b ϕA(x)(ω) := ω (mod Fil (Bcris)) = 0. Zx We claim the following: b Lemma 4.9. Let x ∈ T (A) and ω ∈ W. Then, we have that ϕcris(x)(ω)= 0 if and only if ϕcris(x)(ω)= p A A 2 + 0 (mod Fil (Bcris)). b b b 15 + + + Proof. We first remark that as Bcris ⊂ BdR and the filtration of Bcris is the one induced by the + + + filtration of BdR, it is enough to prove the lemma with Bcris replaced by BdR. ∨ : ( ) + ∗ : ( ) + 2 Let us fix ω ∈ W. Then the maps ω Vp A BdR and ω Vp A BdR/I given by ∨ cris ∗ ∨ + + 2 ω (x) := ϕA (x)(ω) and ω := π ◦ ω , with π: BdR − BdR/I is the natural projection, are nat- Q b + b + urally p-linear maps, GF-equivariant. Here I ⊂ BdR→is the maximal ideal of→BdR, whose powers define the filtration of this ring. Therefore, we can see→ω∨ and ω∗ as elements of the following modules: ∨ ( ) + ∼ + GF ω ∈ HomGF Vp A , BdR = T ⊗Qp BdR G ω∗ ∈ Hom V (A), B+ /I2 =∼ T ⊗ B+ /I2 F GF p dR Qp dR ∨ b where we have denoted by T := Vp(A) . The projection map π induces a natural map b G G π : T ⊗ B+ F − T ⊗ B+ /I2 F 0 b dR dR and the lemma will follow if we show that this map is an isomorp hism. For every n > 2 consider the natural exact sequence→ of rings and ideals

n n+1 + n+1 πn + n 0 − I /I − BdR/I − BdR/I − 0. We tensor this exact sequence with T, over Qp, and obtain the exact sequence of GF-modules: → → → → n n+1 + n+1 1⊗πn + n 0 − T ⊗ I /I − T ⊗ BdR/I − T ⊗ BdR/I − 0. The important remark is that the Hodge-Tate weights of T are 0 and −1, therefore → n n+1 ∼ → ∼ →a b → T ⊗ I /I = T ⊗ Cp(n) = Cp(n) ⊕ Cp(n − 1) , with positive n, n − 1 and non-negative integers a, b. As a consequence the long exact GF continuous cohomology sequence gives the exact sequence: + n+1 GF un + n GF 1 C 0 − T ⊗ BdR/I − T ⊗ BdR/I − H (GF, T ⊗ p(n)) − · · · By Tate’s result [Tat67, Theorem 2], the morphisms un induced by 1 ⊗ πn, are isomorphisms for every n > 2. → → → → > : + k GF + 2 GF For every k 3, let vk T ⊗ BdR/I − T ⊗ BdR/I the map u2 ◦···◦ uk−2 ◦ uk−1. Obviously, v is an isomorphism of K-vector spaces and we have k v G G → GF G T ⊗ B+ F =∼ T ⊗ lim B+ /In F =∼ lim T ⊗ B+ /In =∼ T ⊗ B+ /I2 F , dR − dR − dR dR n>2 n>2 where v is induced by the family (v ) > and therefore it is an isomorphism. ← n n 2 ← cris Now, by the basic properties of ϕA (cf. [Col92, Proposition 3.1.(iii)]), for every ω ∈ W and every n > 0 we have ϕcris(x)(φn (ω)) = φn (ω)= ϕn ω = 0 A D D , x x cris r cris Z + Z n where ϕ := (ϕ ) and ϕ bdenotes the Frobenius on Bcris. Let us denoteby H := n>0 φD(W) ⊂ D. We remark that as the characteristic polynomial of φD annihilates D (and so W), in the sum defining H we can stop for n large enough. Let us remark that ϕ(H)= H and W ⊂PH. Lemma 4.10. We have H = D. • Proof. We first claim that H, (φD)|H, Fil (H) is a weakly admissible filtered, Frobenius module. It certainly is a filtered, Frobenius submodule of D, therefore t (H) > t (H) , by the admissibility N H of D. Moreover, the Hodge polygon of D consists of a horizontal segment of length h − g, where dimF(D) = h and dimF(W) = g and a segment of slope 1 and hight g. As the filtration of H is given by: Fili(H) = H for i 6 0, Fil1(H) = W and Filj(H) = 0 for j > 2, the Hodge polygon of H 16 ′ ′ has a horizontal segment of length h − g, h = dimF(H) and a segment of slope 1 and height g. So tH(H)= tH(D)= g. On the other hand, by [Kat81, Corollary 5.7.7 & 5.7.8], we have an exact sequence of filtered, Frobenius modules 1 1 0 − Het(A, Zp) − Hcris(A/F) − D − 0 where A denotes the special fiber of the Neron model of A. The above exact sequence identifies 1 the first term with the slope →0 submodule of→Hcris(A/F) and→D as→ the slope > 0 quotient of the same Frobenius module. As all the slopes of φD on D (so on H as well) are > 0, we have that tN(H) 6 tN(D) = tH(D) = g. Therefore we get that tN(H) = tH(H) = g, i.e. H is a indeed a weakly admissible filtered, Frobenius submodule of D. Therefore, the quotient D/H (in the category of filtered, Frobenius modules) is weakly admissible. But D/H has positive slopes of Frobenius and its filtration is: Fili(D/H) = D/H for i 6 0 and Filj(D/H) = 0 for all j > 1. So the Hodge polygon of H has only a horizontal segment and as the slopes of Frobenius on D/H are positive, if it is non-zero, it cannot be weakly admissible. So D/H = 0 i.e. D = H. As a consequence of Lemma 4.9 and Lemma 4.10, ϕcris(x)(ω)= 0 for all ω ∈ D, i.e. , ϕcris(x)= A A cris cris cris 0. Since ϕ = (ϕA )|T (A) and ϕA is injective by [Col92, Théorème 5.2.(ii)], we have that x = 0, A p b b and hence our desired result. b b 4.11. A first application of Theorem 3.8. We keep all the notations from the previous sections and Subsection 1.7, i.e. let A be an abelian variety over F satisfying the assumptions there. Let K ⊂ L ⊂ K be such that L is a finite extension of K and denote GL the absolute Galois group of L.

GL G Lemma 4.12. We have that Tp(A) = 0 and per(BA) = (BA) L . Proof. Theorem 3.8 implies that we have the following commutative diagram with exact rows:

ϕA 0 − Tp(A) − Lie(A) ⊗ Cp(1) ∩ ∩ || ϕA 0 − per(BA) −→ BA −→ Lie(A) ⊗ Cp(1)

GL Now we take GL-invariants of this diagram and use the fact that (Lie(A) ⊗ Cp(1)) = 0 and G → → → (per(BA)) L = per(BA). Of course, it follows, either from the lemma, or using the diagram in the proof of the lemma, G G GM that for every ﬁnite extension M of F in K, we have (Tp(A)) M = 0 and (BA) M = per(BA) .

5. Another point of view on the Fontaine integration map In this section, we give another perspective on the Fontaine integration map, which will naturally lead us towards two applications of Theorem A. We keep all the notations from the previous sections and Subsection 1.7. Recall that we let O := O and we have the O-module Ω := Ω1 with its canonical derivation d: O Ω. Note K O/OK that d is surjective and Ω is p-divisible, and let us denote O(1) := ker(d). → (1) (1) Lemma 5.1. Let Ainf denote the p-adic completion of O . Then, the exact sequence of GF-modules d 0 O(1) O Ω 0 induces another exact sequence: → → → → ( ) (1) γ 0 Tp Ω Ainf OCp 0, ( ) (1) where γ is an OF-algebra homomorphism and Tp Ω is seen as an ideal of Ainf of square 0. → → 17 → → Proof. We consider the diagram

0 O(1) O d Ω 0 pn pn pn 0 O(1) O d Ω 0.

The snake lemma gives the exact sequence of GF-modules:

0 Ω[pn] O(1)/pnO(1) O/pnO 0. By taking the projective limit with respect to n of this exact sequence, we obtain the claim. → → → → Remark 5.2. The above proof follows from [Col12, Lemme 3.8] and also from [IZ99, Corollary 1.1]. We now describe another construction of the Fontaine integral restricted to the Tate module of A. Recall that we have the isomorphism Lie(A)(O ) =∼ H0(A, Ω1 )∨. By Lemma 5.1, we have F A/OF the short exact sequence ( ) (1) 0 Tp Ω Ainf OCp 0, ( ) where T (Ω) is an ideal of A 1 such that (T (Ω))2 = 0. p inf → p → → → By deﬁnition, we have

( )( ) ( ) ∼ ( (1)) ( ) Lie A OF ⊗OF Tp Ω = ker A Ainf A OCp , and hence we have the following short exact sequence of abeli→an groups with GF-action ( )( ) ( ) ( (1)) ( ) 0 Lie A OF ⊗OF Tp Ω A Ainf A OCp 0. Consider the following commutative diagram with exact rows → → → → ( )( ) ( ) ( (1)) d ( ) 0 Lie A OF ⊗OF Tp Ω A Ainf A OCp 0 n pn p pn ( )( ) ( ) ( (1)) d ( ) 0 Lie A OF ⊗OF Tp Ω A Ainf A OCp 0.

The snake lemma gives a GK-equivariant map n ∼ n n νn : A(OCp )[p ] = A(K)[p ] Lie(A)(OF) ⊗OF Ω[p ] and by taking the projective limit over n’s, we obtain a map →

ν: Tp(A) Lie(A)(OF) ⊗OF Tp(Ω).

→ Proposition 5.3. The map obtained above

ν : Tp(A) Lie(A)(OF) ⊗OF Tp(Ω) ⊂ Lie(A)(OF) ⊗OF Cp(1) coincides with Fontaine’s integral, i.e. we have ν = (ϕA)| ( ). → Tp A Proof. In [Win94, Section 4, page 394], Wintenberger used a generalization of the above construction to obtain an integration pairing which coincides with the Colmez integration pairing h·, ·iCz. The result now follows from [Col92, Proposition 6.1]. 18 We consider the diagram

( )( ) ( ) ( (1)) d ( ) 0 Lie A OF ⊗OF Tp Ω A Ainf A OCp 0 n pn p pn ( )( ) ( ) ( (1)) d ( ) 0 Lie A OF ⊗OF Tp Ω A Ainf A OCp 0. Above, we only wrote a piece of the snake lemma, and by writing more of it, we have an exact sequence of GK-modules ( (1))[ n] ( )[ n] ( )( ) [ n] 0 A Ainf p A O p Lie A OF ⊗OF Ω p . By taking projective limits, we have the exact sequence → → → ( ( (1))) ( ) ϕA ( )( ) ( ) ( )( ) C ( ) 0 Tp A Ainf Tp A Lie A OF ⊗OF Tp Ω ⊂ Lie A OF ⊗OF p 1 . (5.1) (1) G Therefore, Theorem 4.8 implies that Tp(A(A )) = 0 and Theorem A implies that if Tp(A) K = 0, → → → inf ( ( (1))) = then Tp A Ainf 0. b (1) To study consequences of this property, we will use another ring instead of Ainf . : (1) := Definition 5.4 ([Fon94]). Let θ Ainf OCp denote the projection map. Then, we define Df −1 θ (O). In[Fon94, Remark 1.4.7], Fontaine gives the following construction of Df. Let us recall that → p p p V (Ω)= T (Ω) ⊗ Q = lim Ω Ω . . . Ω... . p p Zp p − We recall that Ω and Vp(Ω) are O-modules. We make R := Vp(Ω) ⊕ O into a commutative ring ← ← ← by defining multiplication as follows: (u, α)(v,←β) = (βu + αv, αβ) for (u, α), (v, β) ∈ R, i.e. we require that Vp(Ω) is an ideal of R of square 0. Then we have

Df = {(u = (un)n>0, α) ∈ R | d(α)= u0}.

By Deﬁnition 5.4, we have an exact sequence of GK-modules θ 0 Tp(Ω) Df O 0, (5.1) (1) where θ(u, α) = α, and the p-adic completion of Df is Ainf . We note that we may construct the → → → → (1) diagram above in the same way using Df and the exact sequence (5.1) instead of Ainf . Instead of the exact sequence (5.1) above, we will have the following exact sequence

ϕA 0 Tp(A(Df)) Tp(A) Lie(A)(OF) ⊗OF Tp(Ω) ⊂ Lie(A)(OF) ⊗OF Cp(1). (5.2)

Again, the Theorem 4.8 implies that Tp A(Df) = 0 and Theorem A implies that Tp A(Df) = 0. This observation→ allows us→ to prove→ that when A satisﬁes Assumption 3.1, the w-topology on A(O) induces the discrete topology on Atorb. First, we need the following lemma. n (1) ′ ′ n ′ Lemma 5.5. Let x ∈A[p ](O ), then there is x ∈A(Df) with θ(x )= x and such that [p ](x )= 0.

Proof. Recall that Df := { (xn)n, y) ∈ Vp(Ω) × O | x0 = dy}, i.e. we have an exact sequence θ 0 Tp(Ω) Df O 0 and this exact sequence splits over O(1) ⊂ O, i.e. the following diagram is cartesian and has exact rows → → → → θ 0 − Tp(Ω) − Df − O − 0 || ∪ ∪ (1) θ (1) 0 −→ Tp(Ω) −→ Tp(Ω) ⊕ O −→ O −→ 0. 19 → → → → (1) In particular, the section s: O − Df is defined by s(x) := (0, x). Then s defines a morphism (1) n (1) n s: A(O ) − A(Df), and if x ∈A[p ](O ) then s(x) ∈A[p ](Df). → Proposition 5.6. Suppose that A satisfies Assumption 3.1. Then the w-topology on A(O) induces the → discrete topology on Ator. (1) Proof. We claim that Ap−tor ∩A(O ) is a finite set, which will imply the lemma for Ap−tor. As (1) Ap ′−tor ⊂ A(OK) the proof in this case follows because A(OK) ⊂ A(O ) and the w-topology (1) on A(O) induces the p-adic topology on A(O ), and hence the discrete topology on Ap ′−tor. To prove original claim, we recall that since A satisfies Assumption 3.1, Theorem A implies that (1) Tp A(Df) = 0. Now Lemma 5.5 implies that Ap−tor ∩A(O ) is finite. 6. A p-adic uniformization result for abelian varieties with good reduction In this section, we prove our Theorem B which states that the p-adic points of an abelian variety A over F with good reduction which satisfies Assumption 3.1, admit a certain type of p-adic uniformization, which has not been observed before and strongly resembles the classical complex uniformization. We recall from Section 1 that there is a certain subgroup A(p)(K) of A(K) which we are interested in studying. We start by recalling the notation Ator(K), Ap−tor(K), Ap ′−tor(K) which denotes the torsion subgroup, the p-power torsion subgroup, and the prime-to-p power torsion subgroup of the group A(K), respectively. We consider the w-topology on A(K), it induces the discrete topology on Ap ′−tor(K), therefore w this subgroup is closed in A (K). Similarly, the w-topology on BA induces the discrete topology on per(BA), and so this subgroup is closed in BA. Definition 6.1. We define the topological abelian groups

(p),w w (p) A := A (K)/Ap ′−tor(K) and BA := BA/per(BA), on which we consider the respective quotient topologies. Remark 6.2. Moreover, we have natural isomorphisms of groups:

(p),w ∼ ∼ (p) ∼ A ⊗Z Q = A(K) ⊗Z Q = A(K)/Ator(K) and BA = BA ⊗Z Q.

As before the w-topology of A(K) induces the discrete topology on Ator(K) (Proposition 5.6), there- w (p),w fore this subgroup is closed in A (K). We deﬁne on A ⊗Z Q the topology induced by the w isomorphisms above, with the quotient topology on A (K)/Ator(K).

We recall from Section 2 that we have an exact sequence of GF-modules

0 Tp(A) BA A(K) 0. (6.1) We consider the following commutative diagram, with exact rows and columns → → → → per(BA) = Ap ′−tor(K) ∩ ∩ 0 − Tp(A) − BA − A(K) − 0 || (p) (p),w 0 −→ Tp(A) −→ BA −→ A (K) −→ 0 ↓ ↓ We mentioned above that we can see A(p) also as a subgroup of A(K), namely there is a natural → → → → section s: A(Cp) Ap ′−tor(K) of the natural inclusion Ap ′−tor(K) ⊂ A(Cp) defined as follows 20 → (see [Fon03]). The exponential is defined locally, i.e. there is an OF-lattice Λ in Lie(A)(OF) and a continuous map : (C ) expΛ OCp ⊗OF Λ A p , ( ( )) = (C ) n ( ) such that logA expΛ x x for x ∈ OCp ⊗OF Λ. For x ∈ A p , for n ≫ 0 we have p logA x ∈ → OCp ⊗OF Λ and n xp s(x) := p−nπ n , expΛ p logA(x) ! where π: Ator(K) Ap ′−tor(K) is the canonical projection. Above, we denoted (following [Fon03]) A(Cp) multiplicatively and Ap ′−tor(K) additively. Then, s is continuous and GK-equivariant. Let s: A(Cp) → Ap ′−tor(K) denote the section to the inclusion Ap ′−tor(K) ⊂ A(Cp) defined (p) (p) (p) above. We denote by A (Cp) := ker(s) and A (K) := A(K) ∩ A (Cp). We remark that we (p) ∼ have natural isomorphisms→ of groups A (K) = A(K)/Ap ′−tor(K), and therefore we could have defined A(p)(K) as the subgroup of A(K) given by A(K) ∩ ker(s). But we believe that it is better, for topological considerations, to think about A(p)(K) as a quotient of A(K) rather then a subgroup. We now continue our investigation of the structure of the topological group A(p),w(K); we will abuse notation and denote by : (p) ( )( ) C ( ) ϕA BA Lie A OF ⊗OF p 1 (p) the restriction of ϕ to B . Given that A satisfies Assumption 3.1, Theorem A implies that ϕ is A A → A injective. We then have a natural commutative diagram of Zp- and GF-modules

(p) (p),w (p) 0 Tp(A) BA A (K)A 0

ϕA ( )( ) C (1) (Lie A OF ⊗OF p ) 0 ϕ (Tp(A)) Lie(A)(OF) ⊗ Cp(1) 0 A OF ϕA(Tp(A)) where above and throughout this section, equality signs in diagrams will denote isomorphisms. We let ֒ p),w) ιA : A (K) (Lie(A)(OF) ⊗OF Cp(1))/ϕA(Tp(A)) denote the injective, GF-equivariant, Zp-linear homomorphism induced by the diagram and the fact that (ϕA) is injective. We observe→ that as ϕA is continuous with respect to the w-topology (p) (p),w on BA , ιA is continuous with respect to the w-quotient topology on A (K) and the quotient topology on Lie(A) ⊗ Cp(1)/ϕA(Tp(A)). Notice that as Tp(A) is compact, ϕA(Tp(A)) is a closed subgroup of Lie(A) ⊗ Cp(1), so the quotient topology makes sense. Next, we will describe the image ιA and then show that the triple ֒ (Tp(A),Lie(A)(OF) ⊗OF Cp(1), ϕA : Tp(A) Lie(A)(OF) ⊗OF Cp(1)) determines the group A(p),w(K). This latter result is some analogue of [SW13, Theorem B]. While we do not obtain an equivalence of categories, we are able→ to show that the above triple determines more than just the p-divisible group of A. Consider the more general situation: let T be a free Zp-module of rank 2g with a continuous GF-action, V a free OF-module of rank g where GF acts trivially on V, and an injective Zp-linear, ֒ GF-equivariant map ϕ: T V ⊗OF Cp(n), with n ∈ Z, n 6= 0. We remark that the existence of ϕ implies T GK = 0. ( C ( )) ( ) We now define a certain→ class of elements in V ⊗OF p n /ϕ T . 21 Definition 6.3. Let x ∈ (V ⊗OK Cp(n))/ϕ(T). We say x is algebraic if the orbit GF · x ⊂ (V ⊗OF Cp(n))/ϕ(T) is finite. If such an x is algebraic, then there is L ⊂ K, [L : F] < such that x ∈ GL (V ⊗OF Cp(n)/ϕ(T)) . ∞ Definition 6.4. Suppose that T is a crystalline GF-representation. Let x ∈ (V ⊗OF Cp(n))/ϕ(T) be an algebraic element. We say that x is a crystalline element if one of the following happens: (1) x is a torsion element, or (2) x is non-torsion, and if we denote

α: V ⊗OF Cp(n) (V ⊗OF Cp(n))/ϕ(T) −1 the natural projection, then we have that α (xZp) is a crystalline GL-representation, for → GL every L with [L : F] < such that x ∈ (V ⊗OK Cp(n))/ϕ(T))) .

Remark 6.5. Let us observe that if T is a crystalline GF representation and x ∈ (V ⊗O Cp(n)) /ϕ(T) ∞ F is a non-torsion, algebraic element, invariant by GL for [L : F] < , we have a natural commutative diagram with exact rows

ϕ α ∞ 0 T V ⊗OF Cp(n) (V ⊗OF Cp(n)) /ϕ(T) 0

ϕ −1 α 0 T α (xZp) xZp 0.

−1 Therefore, α (xZp) ⊗Zp Qp is a Qp-vector space of rank dimZp (T)+ 1 which is a GL-representation. This representation is crystalline if and only if the Galois cohomology class deﬁned by the second 1 row of the diagram tensored with Qp over Zp lives in Hf (L, T ⊗Zp Qp).

Theorem 6.6. Recall the injective, continuous, GF-equivariant, Zp-linear homomorphism ֒ p),w) ιA : A (K) (Lie(A)(OF) ⊗OF Cp(1))/ϕA(Tp(A)).

An element x ∈ (Lie(A)(OF) ⊗O Cp(1)) /ϕA(Tp(A)) lies in the image of ιA if and only if x is crystalline. F →

Proof. As Cp is a field, Cp(1) and (Lie(A)(OF) ⊗OF Cp(1)) are divisible groups. Let us first determine the torsion of (Lie(A)(OF) ⊗OF Cp(1)) /ϕA(Tp(A)). We have the following isomorphisms of Zp and GF-modules: 1 ((Lie(A)(O ) ⊗ C (1)) /ϕ (T (A))) [pn] =∼ ϕ (T (A)) /ϕ (T (A)) F OF p A p pn A p A p 1 =∼ ϕ T (A) /T (A) A pn p p ∼ n = ιA A(K)[p ] . This implies that ∞ ∼ ∼ ∞ ((Lie(A)(OF) ⊗OF Cp(1)) /ϕA(Tp(A))) [p ] = (ϕA ⊗ 1Qp )(Vp(A))/ϕA(Tp(A)) = ιA A(K)[p ] , (p) where we recall that Vp(A) = Tp(A) ⊗Z Q ⊂ BA ⊗Z Q. Moreover, ιA induces a GF-isomorphism between the p-power torsion of the two modules. From this, we deduce that every torsion point of (Lie(A)(OF) ⊗OF Cp(1)) /ϕA(Tp(A)) is algebraic (and, by definition crystalline).

We now compare the non-torsion points. Let x ∈ (Lie(A)(OF) ⊗OF Cp(1)) /ϕA(Tp(A)) be a non-torsion point. We consider its image, which we denote also by x, in (( ( )( ) C ( )) ( ( ))) Lie A OF ⊗OF p 1 /ϕA Tp A ∞ ((Lie(A)(OF) ⊗OF Cp(1)) /ϕA(Tp(A))) [p ] 22 which is isomorphic to (Lie(A)(OF) ⊗OF Cp(1)) /(ϕA ⊗ 1Qp )(Vp(A)). Recall that we have the diagram:

(p) (p),w 0 Vp(A) BA ⊗Z Q A (K) ⊗Z Q 0

ιA (ϕA⊗1Qp ) (ϕA⊗1Qp ) ( )( ) C (1) (Lie A OF ⊗OF p ) 0 (ϕA ⊗ 1Qp )(Vp(A)) Lie(A)(OF) ⊗O Cp(1) 0 F (ϕA⊗1Qp )(Vp(A))

−1 0 (ϕA ⊗ 1Qp )(Vp(A)) α (xQp) xQp 0.

Let L be a ﬁnite extension of F contained in K such that x is GL-invariant, and consider the inclusion

(p),w (p),w GL G (p),w A (L) ⊗Z Q := A (K) ⊗Z Q = A(K) L ⊗Z Q = A(L) ⊗Z Q ⊂ A(K) ⊗Z Q = A (K) ⊗Z Q.

It induces, by pull-back of the ﬁrst row, another commutative diagram with exact rows:

(p) (p),w 0 Vp(A) BA (L) ⊗Z Q A (L) ⊗Z Q 0

ιA (ϕA⊗1Qp ) (ϕA⊗1Q) ( )( ) C (1) (Lie A OK ⊗OK p ) 0 (ϕA ⊗ 1Qp )(Vp(A)) Lie(A)(OK) ⊗O Cp(1) 0 K (ϕA⊗1Qp )(Vp(A))

−1 0 (ϕA ⊗ 1Qp )(Vp(A)) α (xQp) xQp 0.

We wish to consider the long exact, continuous GL-cohomology diagram. The first row is an (p) exact row of finite dimensional Qp-vector spaces and the w-topologies induced by BA ⊗Z Q and (p) respectively A (K) ⊗Z Q on the middle and third terms respectively are the natural p-adic topologies. Therefore this row has a natural, continuous splitting as Qp-vector spaces. The second row is an exact sequence of Qp-Banach spaces, and so these Banach spaces are orthonormalizable. By choosing an orthonormal basis of Lie(A)(OF) ⊗OF Cp(1), indexed by a totally ordered set, such that the first 2g basis elements are the images of a Qp-basis of Vp(A), we obtain a continuous splitting as Qp-Banach spaces of the second row. Therefore we consider the long exact, continuous GL-cohomology of the diagram

∂ 1 0 A(L) ⊗Z Q H (L, Vp(A)

ιA G ( )( ) C (1) L (Lie A OF ⊗OF p ) 1 0 H L, (ϕA ⊗ 1Qp )(Vp(A) 0 (ϕA⊗1Qp )(Vp(A))

γ 1 0 xQp H L, (ϕA ⊗ 1Qp )(Vp(A) where the middle isomorphism follows from [Tat67, Theorem 2]. As we already mentioned, 1 [BK07, Example 3.11] implies that ∂ (A(L) ⊗Z Q)= Hf (L, Vp(A)), and therefore x ∈ ιA (A(L) ⊗Z Q) 1 −1 if an only if γ(x) ∈ Hf (L, Vp(A)). By Remark 6.5, this is equivalent to α (xQp) being a crystalline GL-representation, and hence x being crystalline.

23 Remark 6.7. It follows from the proof of Theorem 6.6 that, for every ﬁnite extension L of F, we have natural isomorphisms:

GL ∼ 1 βL : ((Lie(A)(OF) ⊗OF Cp(1)) /ϕA(Tp(A))) = H (L, ϕA(Tp(A))) , which are compatible with restrictions from L′ to L, if L ⊂ L′.

7. A p-adic uniformization result for the rigid multiplicative group In this section, we study the Fontaine integral for the rigid analytic multiplicative group and deduce a p-adic uniformization result for the rigid analytic multiplicative group, which is similar to that of an abelian variety with good reduction. rig Let F denote a finite, unramified extension of Qp in K and denote G := Gm := Spm(FhT, 1/Ti), with the multiplicative group law, denote the rigid analytic multiplicative group. If L is a sub- × field of K containing F, then G(L) = OL , · , i.e. the multiplicative group of OL. In particular, G(K) = O×, · . In Remark 2.13, we defined the Fontaine integral for an abelian variety with good reduction and we point out that the definitions carry over to the rigid analytic multiplicative group. More precisely, we denote by

φ φ φ B := O× O× ···O× · · · G lim− p where φ(x)= x , and define the map ϕG : BG − Cp(1) via ← ← ← ← ∗ dT dun ϕ (u ) > := (u ) = ∈ V (Ω)= C (1) G n n 0 n T → u p p n>0 n n>0 for all (un)n>0 ∈ BG. rig The first result of this section prove the analogue of Theorem A for G = Gm . × p Theorem 7.1. Let u = (un)n>0 ∈ BG, i.e. a sequence such that un ∈ O and (un+1) = un for all n > 0. Then, ϕG(u) = 0 if and only if u is a periodic sequence, which implies that u0 ∈ µm with (m, p)= 1. We first prove a lemma. × Lemma 7.2. Let u := (un)n>0 ∈ BG be an element such that α(u) := u0 ∈ OL , for some finite extension G L of F in K. If ϕG(u)= 0 then u ∈ B L .

Proof. First, we note that ϕG : Tp(G) Cp(1) is injective and its image is Zp(1) seen as a subgroup −1 −1 of Cp(1). As for any σ ∈ GL we have σ(u)u ∈ Tp(G), we have that ϕG(σ(u)u )= σ ϕG(u) − ϕ (u) = 0. Now using the injectivity of the restriction of ϕ to T (G) we see that σ(u)u−1 = 1, G → G p i.e. σ(u)= u.

Proof of Theorem 7.1. Let u = (un)n>0 ∈ BG and let Kn denote the extension K[un]. By Remark 2.13, the statement of Theorem 7.1 is equivalent to the sequence n + δ(un)− v(un) is bounded if and only if u is not periodic. Indeed, we may take m(n) = n in Remark 2.13, and then the statement ∗ follows from noting that un(dT/T)= d(un)/un and using Lemma 2.4.(6) to compute δ(d(un)/un). First, we prove that if u = (un)n>0 is periodic, then the sequence n + δ(un)− v(un) is unbounded. This follows because Lemma 3.5 tells us that that u is the unique path based at u0, and × since un ∈ OK for all n > 0 by Lemma 3.4, the sequence n + δ(un)− v(un) is unbounded. We now turn to the converse. Let u = (un)n>0 be a sequence as in the statement of Theorem 7.1 which is not periodic. There are two possibilities: either u0 ∈ µm with (m, p) = 1 and there is n0 with un0 ∈ OK\OK, in which case it is easy to see that the sequence n + δ(un)− v(un) is × bounded. The second case is if u0 = zw ∈ OL , for some ﬁnite extension L of F in K, with z ∈ µm 24 for (m, p) = 1 and v(w − 1) > 0 and we will now analyze this case. The hypothesis implies that × the image of u0 ⊗ 1 in OL ⊗Z Q is non-zero. Consider the exact sequence α × 0 Vp(G) BG,L ⊗Z Q − OL ⊗Z Q 0 (7.1) obtained by tensoring with Q the pull-back of the exact sequence → → → → × 1 − Tp(G) − BG − O − 1 × × by the inclusion OL ⊂ O . It is an extesion of ﬁnite dimensional Qp-vector spaces, with continu- → → → → ous action of GL, so the long, exact, continuous GL-cohomology sequence reads:

GL α × ∂ 1 (BG,L ⊗Z Q) − OL ⊗Z Q − H (GL, Vp(G)).

GL GL As B = B and Bloch–Kato [BK07] tell usthat ∂ is injective, Lemma 7.2 implies that ϕG(u) 6= 0, G G,L → → and hence we have our desired result. With this result, we now move onto our uniformization result. Let us recall that we denoted G := Grig, and let Gw(K) denote G(K) = O× with the w-topology. This topology has a base of n (1) × open neighbourhoods of 1 given by {1 + p O }n∈N and for every x ∈ G(K)= O , a base of open n (1) neighbourhoods of x is given by {x(1 + p O )}n∈N. First, we have continuity of the Fontaine integral. Lemma 7.3. The map ψ: Gw(K) Ω given by ψ(x)= dx/x ∈ Ω is continuous for the discrete topology on Ω. Proof. Let ω ∈ Ω and let U := ψ→−1(ω) ⊂ Gw(K). We want to show that U is open. Suppose y ∈ U, i.e. y ∈ Gw(K) and dy/y = ω. Let n ∈ N and u ∈ y(1 + pnO(1)), i.e. u = y(1 + pnv), with v ∈ O(1). We have du/u = dy/y + d(1 + pnv)/(1 + pnv). Moreover, we see that d(1 + pnv) d(pnv) pndv = = = 0, 1 + pnv 1 + pnv 1 + pnv and so du/u = dy/y = ω, i.e. y(1 + pnO(1)) ⊂ U. This implies that U is open.

Next, we show that the w-topology on G(K) induces the discrete topology on the torsion subgroup.

Lemma 7.4. Let Gtor = Gp−tor ⊕ Gp ′−tor ⊂ G(K) denote the torsion subgroup of G(K). Then the w- topology of G(K) induces the discrete topology on Gtor. n (1) −1 n (1) Proof. Let ζ ∈ µp∞ (K) and let x ∈ ζ(1 + p O ) ∩ µp∞ (K). Then y := xζ ∈ 1 + p O , so dy = 0, × (1) ∞ × (1) × i.e. y ∈ O ∩ µp (K) = {1}. So x = ζ. On the other hand Gp ′−tor ⊂ OK ⊂ O ⊂ O and the w-topology of O× induces the p-adic toplogy of O×, which induces the discrete topology on K Gp ′−tor.

The w-topology on G(K) defines the projective limit topology on BG such that the sequence w ∼ w 1 Tp G (K) = Zp(1) BG G (K) 1 (7.1) w is an exact sequence of topological abelian groups. By Lemma 7.4, the topology induced by BG on Zp(1) is the p-adic topology,→ and it follows immediately→ from→ Lemma→ 2.17 that ϕG : BG Cp(1) w is continuous, with the p-adic topology on Cp(1) := Vp(Ω).Moreover, as the GK-action on O is w × w continuous (cf. [IZ99]) it follows that the GK-action on G (K) = (O ) is continuous, and→ since the GK-action on BG is continuous, the exact sequence (7.1) is an exact sequence of continuous GK-representations. 25 Theorem 7.1 asserts that ker(ϕG) = per(BG), the periodic sequences in BG. Also, under the ∼ identification Tp(Ω) ⊗Zp Qp = Cp(1), ϕG(Zp(1)) is identified with Zp(1) ⊂ Cp(1). Therefore we have the following commutative diagram with exact rows and columns: 0 0

per(BG) Gp ′−tor

w 0 Zp(1) BG G (K) 0

ϕG

0 Zp(1) Cp(1) Cp(1)/Zp(1) 0 where the equal signs mean isomorphisms. In particular we have an injective, continuous and GK-equivariant map ֒ p),w w) ιG : G(K) := G (K)/Gp ′−tor Cp(1)/Zp(1).

→ Theorem 7.5. We have that an element x ∈ Cp(1)/Zp(1) belongs to the image of ιG if and only if x is crystalline (Deﬁnition 6.4).

Proof. We remark that the torsion subgroup of Cp(1)/Zp(1) is the subgroup ∼ ∞ Qp(1)/Zp(1) = ιG(G[p ](O)= µp∞ (O).

Therefore ιG induces an isomorphism on the torsion subgroups of the domain and target, so we may look at the map induced by ιG on the quotients of the domain and target by the torsion × .(subgroups i.e., we look at ιG : O ⊗Z Q ֒ Cp(1)/Qp(1 Let x ∈ Cp(1)/Qp(1) and suppose it is algebraic, i.e. invariant under some GL, for L a ﬁnite extension of F in K. Consider the exact→ sequence (7.1) in the context of the diagram of p-adic GL-representations × 0 Vp(G) BG,L ⊗Z Q OL ⊗Z Q 0

ϕG ϕG ϕG

0 Qp(1) Cp(1) Cp(1)/Qp(1) 0.

The GL-continuous cohomology diagram associated to it is × ∂ 1 0 − OL ⊗Z Q − H (GL, Qp(1)) − ∩ιG || ′ GL ∂ 1 0 −→ Cp(1)/Qp(1) −→ H (GL, Qp(1)) −→ 0

1 1 GL As the image of ∂ is H (GL, Qp(1)) ⊂ H (GL, Qp(1)), we see that x ∈ Cp(1)/Qp(1) is in the f → → → image of ι if and only if ∂′(x) ∈ H1(G , Q (1)), i.e. if and only if x is crystalline. G f L p 7.6. A remark on the rigid multiplicative uniformization. Let us recall a few of Fontaine’s definitions: φ φ φ R := O pO O pO O pO · · · lim− / / / p where as usual φ(x)= x . This is an Fp-algebra (i.e. a ring of charactersitic p which is a complete ← ← ← ∼ valuation ring, for a certain rank←1-valuation) defined by Fontaine and it can also be seen as R = Ainf/pAinf. Let us also define φ φ R ′ := lim O O · · · − Cp Cp 26 ← ← ← ′ with the natural, multiplicative map R R given by (a0, a1, . . . ) (a0 (mod p), a1 (mod p), . . . ). Fontaine shows that this map an isomorphism of multiplicative monoids, and we will identify these two monoids. Recall that R is a commutative→ ring of characteristic→ p. × We remark that BG is a subset of R . The inclusion is a group homomorphism but it is not × continuous for the w-topology on BG and the valuation topology on R . Moreover the projection on the first factor induces an exact sequence: 0 − Z (1) − R× − O× − 1 p Cp || ∪ ∪ ∗ 0 −→ Zp(1) −→ BG −→ G(K)= O −→ 1 × Because of topological reasons, Fontaine’s integration ϕG : − Cp(1) does not extend to R , and ι does not extend to O× .→ → → → G Cp → 8. Consequences of Theorem A: information on the ramification of p-power torsion points of an abelian variety. In this section, we will derive some results regarding the ramification properties of the field extensions obtained by adjoining the coordinates of the p-power torsion points of the formal group A of A as above. We work now in the notations of Section 5. We recall that F is a finite, unramified extension of Qbp in K and A an abelian variety with good reduction over F and Néron model A over OF. In that n (1) section we proved that if 0 6= P ∈ A[p ](O ), then there is a Q ∈ A(Df), with θ(Q) = P and such that [pn](Q)= 0. In the next proposition, we show a converse of this property, at least for the formal group A of A. > Proposition 8.1.b Let A denote the formal group of the abelian scheme A and fix n 1 an integer. Let 0 6= P ∈ A(O)[pn]\A(O(1)) b and let Q ∈ A(D ) be a point such that θ(Q)= P. Then [pm]Q 6= 0 for all m > n. f b b Proof. Let m > n and denote by b m [p ](X1, . . . , Xg):=(f1(X1, . . . , Xg), f2(X1, . . . , Xg), . . . , fg(X1, . . . , Xg)) m the multiplication by p on A. Let S := OF[[X1, . . . , Xg)]]/I, where I is the ideal generated by m f1, f2, . . . , fg, then we know S is a finite flat OF-algebra, so free OF-module and A[p ] := Spec(S) b with the co-multiplication of A, is a finite flat group-scheme,so A×OF Spec(F) is an étale, therefore smooth, group-scheme over F. This implies that the image in S ⊗OF F of the determinantb of the matrix: b b ∂(f1) ∂(f1) . . . ∂(f1) ∂(X1) ∂(X2) ∂(Xg) . . . .  . . .. .  ∂(f ) ∂(f ) ∂(f )  g g . . . g   ∂(X1) ∂(X2) ∂(Xg)  is a unit.   g n n (1) Let now P = (x1, x2, . . . , xg) ∈ mO ∈ A[p ](O)\A[p ](O ), i.e. there is 1 6 i 6 g such that (1) ′ ′ xi is not in O . Let P = (y1, y2, . . . , yg) ∈ A(Df) such that θ(P ) = P, i.e. yj = αj + xj, with αj ∈ Vp(Ω) and d(xj)= αj,0, for all 1 6 j 6b g. By theb above assumption αi 6= 0. Asin Df we have m ′ αjαk = 0 for all 1 6 j, k 6 g, the Taylor formulab implies that if [p ](P )= 0 we must have: g g ∂(f ) ∂(f ) f (x , . . . , x )+ s (x , . . . , x )α = s (x , . . . , x )α = 0 s 1 g ∂(X ) 1 g j ∂(X ) 1 g j = j = j Xj 1 Xj 1 27 for every 1 6 s 6 g. But the determinant of the matrix

∂(f1) ∂(f1) ∂(f1) (x1, . . . , xg) (x1, . . . , xg) . . . (x1, . . . , xg) ∂(X1) ∂(X2) ∂(Xg) . . . .  . . .. .  ∂(fg) ∂(fg) ∂(fg)  (x1, . . . , xg) (x1, . . . , xg) . . . (x1, . . . , xg)   ∂(X1) ∂(X2) ∂(Xg)    is a unit in K, i.e. it is non-zero and αi 6= 0. This is a contradiction. m Remark 8.2. We note that the group-scheme A[p ] is not smooth over OF (for example S/pS could m m have nilpotents). We also remark that as A[p ] ×OF Spec(F) is smooth, the map θ ⊗ 1: A[p ](Df ⊗OF m b K) A[p ](F) is surjective, but this is clear as Df ⊗OF F = Vp(Ω) ⊕ F. b b We observe that Lemma 5.5 and Proposition 8.1 imply that the map θ gives an isomorphism b →n ∼ n (1) A[p ](Df) = A[p ](O ), for all n > 1. Therefore we have proved:

Theorem 8.3 (=Theorem C). Let A be an abelian variety over F with good reduction. Then there is n0 > 1 b b m (1) such that for every m > n0 and 0 6= P ∈ A[p ](O), we have P ∈/ A(O ). We remark that the above is a result regarding ramification properties of the p-power torsion b b points of our abelian variety with good reduction over F. The integer n0 in the statement is difficult to pin down, but below, we give a sufficient criterion for n0 = 1 after base changing A to K.

8.4. A theorem on the ramification type of the field obtained by adjoining a p-torsion point of the formal group of an abelian variety. In this subsection, we give a criterion for when the integer n0 Theorem 8.3 is equal to 1. Moreover, the main result of this section is a theorem on the ramification type of the field obtained by adjoining the coordinates of a p-torsion point of the formal group of an abelian variety, which we believe to be of independent interest. In this subsection K denotes the completion of the maximal unramified extension of Qp in an algebraic closure of Qp, which we denote K, A will be an ableian variety over a finite extension of Qp, F, contained in K, with good reduction, which we base-change to K. We denote A the Néron model of A, seen as an abelian scheme over Spec(OK). Let A denote the formal group of the abelian scheme A of dimension g over Spf(OK). b Definition 8.5. Consider the multiplication-by-p map

[p](X1, . . . ., Xg) = (f1(X1, . . . ., Xg), f2(X1, . . . ., Xg), . . . , fg(X1, . . . ., Xg)) on A where each fi(X1, . . . ., Xg) is a power series in with coefficients in OK. For each 1 6 i 6 g, define Fi(X1, . . . , Xg) to be the form comprised of monomials of fi which have unit coefficient and minimalb degree, where we consider each monomial X1, . . . , Xg to be of degree 1. Let d1, . . . , dg denote the degree of these forms F1(X1, . . . , Xg), . . . , Fg(X1, . . . , Xg), respectively, which we note are (possibly distinct) powers of p. Let G1(X1, . . . , Xg), . . . , Gg(X1, . . . , Xg) denote the reductions modulo p of the forms F1(X1, . . . , Xg), . . . , Fg(X1, . . . , Xg). Consider the system of equations

G1(X1, . . . , Xg)= G2(X1, . . . , Xg)= · · · = Gg(X1, . . . , Xg)= 0, (8.1)

We say that the formal group A is strict if d1 = d2 = · · · = dg and the only solution to (8.1) is g (0, 0, . . . , 0) ∈ (Fp) . b Remark 8.6. If A is an abelian scheme of dimension 1, then it is clear that A is strict since we will ph have that F1(X1) = uX1 , where h is the height of A. If A is an abelian scheme which can be written as the product of 1-dimension abelian schemes, then again A is strict.b 28 b b Remark 8.7. We can given an equivalent characterization of strict as follows. Consider the g × g matrix M = (aij) where the entry aij consists of the coefficient of Xi in the linear form Gj for each 1 6 i, j 6 g. The condition that A be strict is equivalent to the determinant of M being non-zero. We refer the reader to [dR11, Remark 4.14] for an example of a 2-dimensional formal group of height 4 where this conditionb holds, and here we note that the above degrees all equal p2. Moreover, we remark that the proof from loc. cit. holds for any g-dimensional formal group of 2 height 2g where the degrees d1 = d2 = · · · = dg all equal p . With this definition, we can state our main result. Theorem 8.8. Let A denote the formal group of the abelian scheme A of dimension g. Suppose that A is ∼ strict. For 0 6= P ∈ A[p](O), the field of definition K(P)/K is tamely ramified and OK(P) = OK[x1, . . . , xg]. b b Proposition 8.9. Letb A denote the formal group of the abelian scheme A. Suppose that A is strict. For every 0 6= P ∈ A(O)[p], the coordinates of P are not all in O(1). b b Proof. Let 0 6= P = (x , . . . , x ) ∈ A[p](O) be a non-zero p-torsion point. By Theorem 8.8, we b 1 n know that the extension K(P)/K is tamely ramified and that there exists some coordinate xi which is a uniformizer for K(P)/K. By[Ser79b , Proposition III.6.13], we have that v(∆K(P)/K) > 0 where ∆K(P)/K is the different ideal of K(P)/K. Now since xi is a uniformizer for K(P)/K, we have that

δ(xi)=−v(∆K(P)/K) < 0, (1) where δ is the function deﬁned in Deﬁnition 2.3. By Lemma 2.4.(5), xi ∈/ O as desired. For the remainder of this section, we focus on proving Theorem 8.8. The proof can be broken down into three steps.

(1) Given a non-zero p-torsion point P = (x1, . . . , xg) ∈ A(O)[p], we will carefully construct ∗ linear combinations zi of the x1, . . . , xn which satisfy nice properties in terms of their valuations and distances between their K-conjugates. See Lemmab 8.10. (2) Next, we consider the change of variables (i.e., the isomorphism of formal groups) which ∗ sends the coordinate Xi to the linear combination Zi described above. We use the prop- ∗ ∗ erties of the zi and the strictness of A to precisely determine the valuation of zi and to estimate the valuation of the difference between them and their K-conjugates. (3) Finally, we use Krasner’s lemma to deduceb that one of the original coordinates x1, . . . , xg of P must be a uniformizer for the maximal order of K(P), from which Theorem 8.8 follows. Lemma 8.10. Let A denote the formal group of the abelian scheme A of dimension g. Let 0 6= P = ∗ ∗ (x1, . . . , xg) ∈ A[p](O). There exist linear combinations z1, . . . , zg of x1, . . . , xg with coefﬁcients in × (OK) ∪ {0} which satisfy:b ∗ ∼ b (1) K(zi ) = K(P), ∗ (2) v(zi )= min{v(x1), . . . , v(xg)}, ∗ ∗ ] ] (3) v(zi − σ(zi )) = min{v(x1 − σ(x1)), . . . , v(xg − σ(xg))} for all σ ∈ Gal(K(P)/K) where K(P) is the Galois closure of K(P), ∗ ∗ t t and such the matrix M representing the change of coordinates (z1, . . . , zg) = M(x1, . . . , xg) is invertible. Here the exponent t indicates the transpose of a matrix. Proof. Let e := [K(P): K]. Our proof will involve making a series of linear combinations. To begin, ∗ we will construct the element z1. First, consider all the linear combinations of the form × B1 := {z = u1x1 + · · · + ugxg where ui ∈ (OK) ∪ {0} and u1 6= 0}. (8.1)

By our assumptions on K, the set of ui(mod p) is infinite, with ui as in the above formula, and hence we may find one linear combination, call it z1, satisfying the following two conditions: 29 (1) v(z) > v(z1) for all other linear combinations z from B1, ∼ (2) K(z1) = K(P). To show that condition (2) holds, consider the following. There are exactly e embeddings of K(x1, . . . , xg) into the fixed algebraic closure K, call them σ1, . . . , σe. Note that the vectors wj := (σj(x1), . . . , σj(xg)), 1 6 j 6 e, are distinct. Indeed, if for some i 6= j the vectors wi and wj coincide, then σi and σj will coincide at x1, . . . , xg and so they will coincide on K(x1, . . . , xg), which is not the case. Consider now, for each pair (i, j) with 1 6 i, j 6 e and i 6= j, the hyperplane Hi,j given by g

Hi,j = (c1, c2, . . . , cg): c1, . . . , cg ∈ K, cl(σi(xl)− σj(xl)) = 0 . = Xl 2 Since the vectors wj, 1 6 j 6 e are distinct, none of Hi,j covers the full space. Denote by H the union of these ﬁnitely many hyperplanes. Choose now any c1, . . . , cg ∈ K such that the point (c1, c2, . . . , cg) lies outside H. Then we claim that the element

z := c1x1 + · · · + cgxg satisfies K(z) = K(x1, . . . , xg). Indeed, σ1(z), . . . , σe(z) are distinct. For, if two of them are equal, say σi(z) = σj(z) with i 6= j, then (c1, c2 . . . , cg) is forced to lie in Hi,j. Thus σ1(z), . . . , σe(z) are distinct, so z has at least e distinct conjugates over K. Hence [K(z): K] > e and in conclusion K(z)= K(x1, . . . , xg) i.e., z1 has degree e over K. t We pause to note that the matrix M representing the change of coordinates (z1, x2, . . . , xg) = t M(x1, x2, . . . , xg) is invertible. Indeed, the matrix M has units along the diagonal, the coefficients of the linear combination z1 in the first row, and zeros elsewhere, hence the determinant is a unit. We now look at the distances between these linear combinations and various of their conjugates over K. Fix σ ∈ Gal(Qp/K) and consider the infimum of the values v(z − σ(z)) for the linear combinations z as in (8.4), i.e. we look at inf{v (z − σ(z)) | z ∈ B1}. We first show that this infimum exists and is attained by some linear combination. Note that if σ|K(z1) = id, then since all linear combinations belong to K(x1, . . . , xg), we have that z − σ(z) = 0 for all linear combinations z, i.e. { ( − ( )) | } = = ( − ( )) = inf v z σ z z ∈ B1 v z1 σ z1 . If we consider σ such that σ|K(z1) 6 id, then we at least have that v(z − σ(z)) < for some linear combinations z, for example for z = z1. We have that the set of possible valuations∞ is discrete, and the set of values v(z − σ(z)) has a lower bound, namely min{v(x1 − σ(x1)), . . .∞, v(xn − σ(xg))}. Thus, the infimum of the set of values v(z − σ(z)) is attained by some linear combination z from (8.4), but note it need not be obtained by z1. ] ] Let G denote Gal(K(P)/K) where K(P) is the Galois closure of K(P)/K. For a fixed σ ∈ G, let zσ ∈ B1 denote one such linear combination attaining the minimum v(zσ − σ(zσ)) = min{v (z − σ(z)) | z ∈ B1} . We claim that we can find a linear combination of z1 and of all of these zσ where σ ∈ G which simultaneously achieves these minima. To do this, consider all linear combinations

∗ × z = z1 + uσzσ where uσ ∈ (OK) ∪ {0}. (8.2) σX∈G ∗ We will choose the uσ’s such that each such z will live in B1. To achieve our desired simultaneous minima, we start with one σ ∈ G, call it σ1. First, we let ∗ z = z1. This linear combination might work in that it already attains the minimum at σ1; by this we mean that v(z − σ1(z)) > v(y1 − σ1(y1)) holds for all linear combinations z ∈ B1 from (8.4). If this is the case, then we set y1 := z1. Now suppose that z1 does not attain the minimum at σ. In × ∗ × this case, we may use any unit u ∈ (OK) and set z := z1 + uzσ1 . Indeed, for any unit u ∈ (OK) and z∗ above, we have that ∗ ∗ v(z − σ1(z )) = v (z1 − σ1(z1)+ u(zσ1 − σ1(zσ1 )) = v(zσ1 − σ1(zσ1 )), 30 × because v(z1 − σ1(z1)) > v(zσ1 − σ1(zσ1 ). Let y1 := z1 + uzσ1 with unit u ∈ (OK) such that y1 ∈ B1. We have that for such y1 ∈ B1, v(y1 − σ1(y1)) 6 v(z − σ1(z) for all z ∈ B1 and that the ′ u s with y1 ∈ B1 have the property that u (mod p) avoids a finite number of elements in Fp. For another automorphism σ2 ∈ G, we proceed along the same lines, that is: if y1 has the property that v(y1 − σ2(y1)) 6 v(z − σ2(y1)) for all z ∈ B1 we set y2 := y1. If the above is not true, ∗ × ∗ ∗ let z := y1 + uxσ2 , for some u ∈ OK . Then as above we have: v(z − σ2(z )) = v (xσ2 − σ2(xσ2 )) ∗ ′ ∗ therefore z realizes the minimum for σ2, for all u s for which z ∈ B1. What about for σ1. The worst that can happen is that v(y1 − σ1(y1)) = v (xσ2 − σ1(xσ2 )), i.e.if wedenoteby π a uniformizer ( ) − ( ) = α − ( ) = α × ∗ − of K P , we have y1 σ1 y1 aπ xσ2 σ1 xσ2 bπ , with a, b ∈ OK(P). Therefore z ∗ a × σ1(z ) = (a + ub)π . Now the residue field of OK(P) is k, therefore by choosing u ∈ OK such that ∗ ∗ ∗ a + ub( mod π) 6= 0 we have v(z − σ1(z )) = v(y1 − σ1(y1)) and therefore y2 := z realizes the minima for both σ1 and σ2. Continuing in this fashion, we arrive at the conclusion that there exist linear combinations of the form ∗ z1 = z1 + uσzσ (8.3) σ∈G × X where uσ ∈ (OK) ∪ {0} which satisfy the following four conditions: ∗ (1) z1 ∈ B1, ∗ ∼ (2) K(z1) = K(P), ∗ (3) v(z1)= min{v(x1), . . . , v(xn)}, ∗ ∗ (4) v(z1 − σ(z1)) = min{v(x1 − σ(x1)), . . . , v(xn − σ(xg))} for all σ ∈ G, as desired. Again, we pause to note that the matrix M representing the change of coordinates ∗ t t (z1, x2, . . . , xg) = M(x1, x2, . . . , xg) is invertible. Indeed, the matrix M has units along the diag- ∗ onal, the coefficients of the linear combination z1 in the first row, and zeros elsewhere, hence the determinant is clearly a unit. We now wish to iterate this construction as follows. First, consider the set of all linear combinations ′ ∗ × B2 := {z = u1z1 + u2x2 + · · · + ugxg where ui ∈ (OK) ∪ {0} and u2 6= 0}. (8.4) Then, we can repeat the above construction to arrive at a linear combination z2 ∈ B2 satisfing: ′ (1) v(z ) > v(z2) for all other linear combinations z from B2, ∼ (2) K(z2) = K(P). Furthermore, we can follow the above construction to say that there exist linear combinations of the form ∗ z2 = z2 + uσzσ (8.5) σ∈G × X where uσ ∈ (OK) ∪ {0} which satisfy the following four conditions: ∗ (1) z2 ∈ B2, ∗ ∼ (2) K(z2) = K(P), ∗ (3) v(z2)= min{v(x1), . . . , v(xn)}, ∗ ∗ (4) v(z2 − σ(z2)) = min{v(x1 − σ(x1)), . . . , v(xg − σ(xg))} for all σ ∈ G, ′ ∗ ∗ t ′ ∗ t Note that the matrix M representing the change of coordinates (z1, z2, . . . , xg) = M (z1, x2, . . . , xg) ′ ∗ is invertible. Indeed, M has units on the diagonal, the coefficients of the linear combination z2 in the second row, and has zeros everywhere else. Although this matrix is not triangular, we can make it so by switching the second row with the first and interchanging the first and second columns; these operations will not change the determinant. After these operations, the matrix becomes triangular with units on the diagonal, and hence the will be invertible. Moreover, we see ′′ ∗ ∗ t ′′ t that the matrix M representing the change of coordinates (z1, z2, . . . , xg) = M (x1, x2, . . . , xg) is invertible since M′′ = M′ · M. 31 We continue in this fashion for all of the remaining coordinates x3, . . . , xg and arrive at our ∗ ∗ desired claim, namely that there exists linear combinations z1, . . . , zg of x1, . . . , xg with coefficients × in (OK) ∪ {0} which satisfy: ∗ ∼ (1) K(zi ) = K(P), ∗ (2) v(zi )= min{v(x1), . . . , v(xg)}, ∗ ∗ (3) v(zi − σ(zi )) = min{v(x1 − σ(x1)), . . . , v(xg − σ(xg))} for all σG ∗ ∗ t t and such the matrix M representing the change of coordinates (z1, . . . , zg) = M(x1, . . . , xg) is invertible.

We now complete the proof of Theorem 8.8.

Proof of Theorem 8.8. Fix 0 6= P = (x1, . . . , xg) ∈ A[p](O) and let e = [K(P): K]. We first remark that since K is completion of the maximal unramified extension, we have that e > 1, and hence the extension [K(P): K] is totally ramified. Indeed, thisb follows from that fact that the group-scheme A[p] ∗ is connected. In Lemma 8.10, we constructed linear combinations zi of x1, . . . , xg satisfying certain ∗ properties. Let Zi denote the same linear combinations of the coordinates X1, . . . , Xg (so if we wereb ∗ ∗ to evaluate Zi at (x1, . . . , xg) we would recover zi ). The last condition from Lemma 8.10 implies ∗ ∗ that the change of variables (X1, . . . , Xg) 7 (Z1, . . . , Zg) is an isomorphism of formal groups. ∗ ∗ We claim that the isomorphism of formal groups (X1, . . . , Xg) 7 (Z1, . . . , Zg) will preserve strict- ∗ × ness. As each of the Zi are linear combinations→ of X1, . . . , Xg with coefficients in (OK) ∪ {0}, this isomorphism of formal groups will act linearly on terms of minimal→ degree, and hence it changes F1, . . . , Fg by a linear transformation, which is invertible. We note that it also does the same to G1, . . . , Gg, therefore, it transforms the set of solutions of the system by an invertible transformation. Moreover, the system having or not having a single solution (0, . . . , 0) is the same before or after an isomorphism. We pause to note that it is crucial that the degrees d1, d2, . . . , dg from Definition 8.5 are all equal For the remainder of the proof, we work with this isomorphic formal group with coordinates ∗ ∗ ∗ ∗ g (Z1, . . . , Zg). We note that the vector (z1, . . . , zg) will reduce mod p to the point (0, . . . , 0) ∈ k ∗ ∗ because all zi have valuations strictly positive. But the zi have the same valuation so we can ∗ ∗ ∗ ∗ divide all of them by one of them, and consider the vector (z1 /zg, . . . , zg−1/zg, 1), which will not reduce the zero vector over the residue field. Since A was assumed to be strict, the reduction of ∗ ∗ ∗ ∗ (z1 /zg, . . . , zg−1/zg, 1) cannot be a common root of all of G1, . . . , Gg. Therefore, there exists an ∗ ∗ index j for which Gj(z1 /zg, . . . , 1) is not zero in theb residue field k, and hence the valuation of ∗ ∗ Fj(z1, . . . , zn) equals the valuation of each of its individual monomials. ∗ ∗ We now want to determine the valuation of zj , and hence the valuation of every other zi as they ∗ ∗ have the same valuation. By considering the equation fj(z1, . . . , zg), we have ∗ ∗ ∗ 0 = pzj + p(terms of degree between 2 and dj − 1)+ Fj(z1, . . . , zn) + (higher degree terms). ∗ We claim that v(zj )= 1/(dj − 1) where dj is the degree of Fj. To see this choose a unit u1 in OK ∗ ∗ that is a representative for the element in the residue field corresponding to z1 /zg, and similarly choose u2, . . . , ug−1. Then Fj(u1, . . . , ug−1, 1) is a unit in OK, because its image in the residue field is nonzero, by the above choice of j. But this is a form (of degree dj) so we can divide by uj inside Fj, and re-denoting the uj’s in consideration, we have that Fj(u1, . . . , uj−1, 1, uj+1, . . . , ug) is a unit, and moreover,

∗ ∗ ∗dj Fj(z1, . . . , zg)= Fj(u1,...1,...ug)zj + (terms of strictly larger valuation). Plugging this into the the above equation, we arrive at the equation

∗ ∗dj 0 = pzj + Fj(u1, . . . , 1, . . . , ug)zj + (terms of strictly larger valuation). (8.6) 32 The minimum valuation in the equality of (8.6) must be attained in at least two terms, and these ∗ ∗dj terms are forced to be pzj and Fj(u1, . . . , 1, . . . , ug)zj . Our claim now follows since Fj(u1, . . . , 1, . . . , ug) is a unit in OK. ∗ ∗ We now want to study the relationship between the valuations v(zj − σ(zj )) where σ ∈ G. Let u := Fj(u1, . . . , 1,...ug) which is a unit in OK. For each σ ∈ G, we will consider (8.6) and

∗ ∗ dj 0 = pσ(zj )+ uσ(zj ) + (∗ ∗ ∗) (8.7) where (∗ ∗ ∗) corresponds to terms strictly larger valuation. If we subtract equality (8.7) from (8.6), we arrive at the following:

dj ∗ ∗ ∗ ∗ dj 0 = p(zj − σ(zj )) + u(zj − σ(zj ) ) + (interesting terms). (8.8) By condition (4) of Lemma 8.10, the valuation of the “interesting terms" from (8.8) will be larger ∗ ∗ ∗m1 ∗m2 ∗mg than v(zj − σ(zj )). Indeed, theseinteresting terms are in fact monomials of the form z1 z2 ...zn where some of the mi could be zero and the total degree is strictly greater than dj. In any case, we may deal with the difference of such monomials and their conjugates as follows. Suppose for ∗m2 ∗m3 ∗m2 ∗m3 example, that we have a term of the for z2 z3 − σ(z2 z3 ). Then by adding and subtracting ∗m3 ∗m3 ∗ ∗ the term z2 σ(z3 ), the difference we need to deal with can then be written as (z2 − σ(z2)) times ∗ ∗ something of positive valuation, plus (z3 − σ(z3) times something of positive valuation, and so we can use property (4) of Lemma 8.10 for this particular σ to get our desired claim. We now arrive at the crucial claim of the proof. Recall that G denotes the Galois group of ∗ ∗ ∗ the Galois closure of K(P)/K. We claim that v(zj − σ(zj )) = v(zj ) for all σ ∈ G. Assume that ∗ ∗ ∗ v(zj − σ(zj )) = v(zj )+ t where t > 0. By the above discussion and condition (4) of Lemma 8.10, we can save the value t from each term, and since there must be at least two terms of equal valuation in (8.8), we have that

dj ∗ ∗ ∗ ∗ dj v(p(zj − σ(zj ))) > v(u(zj − σ(zj ) )). Note that we cannot guarantee the equality of these valuations because there could be other terms dj ∗ ∗ dj of total minimal degree other than u(zj − σ(zj ) ), but the above inequality will sufﬁce. Using ∗ ∗dj ∗ ∗ the above inequality and previous equality v(pzj ) = v(uzj ), and letting w = σ(zj )/zj , we have that ∗ ∗ ∗ ∗ ∗ ∗ dj t = v(zj − σ(zj )) − v(zj )= v((zj − σ(zj ))/zj )= v(1 − w) > v(1 − w ). Let y = 1 − w. We have that v(y) = t, which by assumption is strictly greater than 0. We now arrive at a contradiction by considering the above inequality and the equation

dj dj 2 dj 1 − w = 1 − (1 − y) = djy + (terms times y )+ y , and noting that all terms in the above have valuation strictly greater than v(y)= t. Therefore, we ∗ ∗ ∗ have that t = 0, and hence v(zj − σ(zj )) = v(zj ) for all σ ∈ G. To conclude our proof, we use Krasner’s lemma [Ser79, Exercise II.2.1] to explicitly describe the ∗ ∗ ∗ extension K(zj )/K and show that [K(zj ): K]= dj − 1. Recall that our zj satisﬁes the equation (8.6). dj−1 ∗ Consider the polynomial P(Z) = p + uZ where u = Fj(u1,...1,...un) as above. Note that zj is ∗ ∗ not a root of P(Z), but it satisﬁes the following inequality v(P(zj )) > v(pzj ) where the right side ∗dj here is also equal to v(uzj ).

On the other hand, the roots θ1, . . . , θdj−1 of P(Z) each have valuation exactly 1/(dj − 1) since P(Z) is Eisenstein at p. We now compute the valuation of the derivative of P(Z) evaluated at a ′ root in two different ways. First, P (θl)= u i6=l(θl − θi) and hence

′ Q dj − 2 v(P (θl)) = v(θl − θi) > . dj − 1 Xi6=l 33 ′ dj−2 Second, we directly compute that P (θl) = (dj − 1)uθl , which yields

′ dj − 2 v(P (θl)) = . dj − 1

The ﬁrst inequality and the second equality imply that v(θl − θi) = 1/(dj − 1) for all 1 6 i 6= j 6 ∗ ∗ ∗ dj − 1. We also have that P(zj )= u(zj − θ1) · · · (zj − θdj−1) and hence

dj−1 ∗ ∗ v(P(zj )) = v(zj − θi). = Xi 1 ∗ There are exactly dj − 1 terms here and since v(P(z )) > 1, it follows that at least one term must ∗ be strictly larger than 1/(dj − 1). Without loss of generality, we may assume that v(zj − θ1) > 1/(dj − 1). Now we have the strict inequality ∗ v(zj − θ1) > 1/(dj − 1)= v(θ1 − θi) ∗ for all 1 v(zj − σ(zj )) = 1/(dj − 1), ∗ and so we can apply Krasner’s lemma again to deduce that K(zj ) ⊆ K(θ1). Therefore, we have ∼ ∗ ∼ shown that K(P) = K(zj ) = K(θ1) and this extension is totally ramified of degree dj − 1, hence ∗ tamely ramified. We also have that zj , which is a linear combination of x1, . . . , xg with unit co- ∼ efficients, is a uniformizer for K(P) and hence OK(P) = OK[x1, . . . , xg]. Finally, we note that there ∗ exists some coordinate xi of P which has valuation v(xi) = v(zj ) by condition (3) of Lemma 8.10 and hence xi is a uniformizer for K(P) as well.

APPENDIX A. Another proof of the injectivity of the Fontaine integral

Let, as before, F be a finite, unramified extension of Qp, K the maximal unramified extension of F in K and GF, GK the absolute Galois groups of F and respectively K. Let A be an abelian variety with good reduction over F, A its Néron model over Spec(OF) and Tp(A) its p-adic Tate module. In this Appendix we present two proofs of the following theorem. G Theorem A.1. If A satisfies the property Tp(A) K = 0, then the Fontaine integral ϕA : Tp(A) − Lie(A)(K) ⊗K Cp(1) is injective. As K is a discretely valued field, which is not complete, we have to first complete it, say we→ denote M the completion of K seen as a subfield of Cp (which denotes, as before, the completion of K). Let M denote the algebraic closure of M in Cp, then M is an algebraically closed field containing K. Moreover, a simple application of Krasner’s lemma (see also [IZ95]) implies that M ∩ K = K. Therefore, if we denote GM := Gal(M/M), we have a natural, continuous group homomorphism induced by restriction: γ: GM GK. Lemma A.2. γ is an isomorphism. → ∼ Proof. We have natural isomorphisms induced by restriction, namely α: Autcont(Cp/K) = GK and ∼ −1 β: Autcont(Cp/M) = GM. Moreover γ = α ◦ β .

Let us now denote by AM the base-change of the abelian variety A to M. Let us observe that if Tp(AM) is the p-adic Tate module of AM, as A is deﬁned over F and K ⊂ M, we have the equality Tp(AM)= Tp(A) as Zp-modules and GM-modules, where the action of GM on Tp(A) is via γ. From now on we will identify these two Tate modules. 34 1 1 On the otherhand, we have the O -module Ω := Ω and the O -module ΩM := Ω K OK/OK M OM/OM and a natural OK-linear map: f: Ω ΩM induced by the inclusions OK ⊂ OM and OK ⊂ OM. Lemma A.3. 1 ⊗ f: O ⊗ Ω Ω O G G The morphism OM M OK M is a M, M-equivariant (the action of M is semi-linear) isomorphism. → → Proof. Both modules are generated over OM by the family

dζn , ζ n n>1 with unique relations dζ dζ p n+1 = n ζ ζ n+1 n n n > 1 (ζ ) > p 1 K 1 ⊗ f for all where n n 1 is a compatible family of -th roots of in . This implies that OM is an isomorphism.

We denote by ϕAM : Tp(AM) Lie(AM)(M) ⊗M Cp(1) Fontaine’s integration map for AM, where here Cp = Vp(ΩM)= Vp(Ω). Note that we have the following diagram: ϕ → AM Tp(AM) − Lie(AM)(M) ⊗M Cp(1) || =∼ ϕA Tp(A) −→ Lie(A)(K) ⊗K Cp(1) It is obvious, given the deﬁnitions and Lemma A.2 and↑ Lemma A.3, that this diagram is commu- → tative, and hence to prove the theorem it is enough to prove that ϕAM is injective. ur Therefore we can re-denote things as follows: let K be the completion of Qp , K an algebraic closure of K, Cp the completion of K and A an abelian variety with good reduction over K satisfying G Tp(A) K = 0.

Theorem A.4 (Yeuk Hay Joshua Lam, Alexander Petrov). In the notations above, if ϕA : Tp(A) Lie(A)(K) ⊗K Cp(1) denotes Fontaine’s integration map, then ϕA is injective. The proof of Theorem A.4 presented below was sent to us, independently, by Yeuk Hay Joshua→ Lam and Alexander Petrov. By the above remarks Theorem A.4 implies Theorem A.1.

Proof. Let S := ker(ϕA ⊗Zp Qp). It sufﬁces to show that S = 0. As Vp(A) is a crystalline representation of GK, S is a crystalline representation with respect to the same Galois group. Consider the following commutative diagram with exact rows:

ϕA 0 − S − Vp(A) − Lie(A) ⊗K Cp(1) ∩ ∩ || ϕA⊗1 0 −→ S ⊗Qp Cp −→ Vp(A) ⊗Qp Cp −→ Lie(A) ⊗K Cp(1) − 0 =∼ || || g ϕA⊗1 0 −→ (Cp) −→ Vp(A) ⊗Qp Cp −→ Lie(A) ⊗K Cp(1) −→ 0 ↓ It follows from Tate’s result [Tat67, Theorem 2] that S is a p-adic representation of GK, which is Hodge-Tate with→ all Hodge-Tate weights→ 0. By[Sen81, Corollary],→ this implies that→ the image of the

Galois representation ρS : GK − EndQp (S) is finite. Let then L/K be the finite, Galois extension, obviously totally ramified, such that ρS factors through ρS : Gal(L/K) − EndQp (S). We wish to show that S = 0.→ As S is a crystalline GK-representation we have G G Gal(L/K) → Gal(L/K) K L GL Dcris(S) = S ⊗Qp Bcris = S ⊗Qp Bcris = M ⊗Qp (Bcris) = Gal(L/K) M ⊗Q p K. 35 The last equality follows from [Fon82, Proposition 5.1.2]. Therefore, from the above we have Gal(L/K) Gal(L/K) GK dimQp (S)= dimK(Dcris(S)) = dimQp (S ). In other words, we have that S = S = S . G But S ⊂ Vp(A) and Vp(A) K = 0, therefore, S = 0.

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37 ADRIAN IOVITA,CONCORDIA UNIVERSITY,DEPARTMENT OF MATHEMATICS AND STATISTICS,MONTRÉAL,QUÉBEC, CANADA AND DIPARTIMENTO DI MATEMATICA, UNIVERSITA DEGLI STUDI DI PADOVA, PADOVA, ITALY Email address: [email protected]

JACKSON S. MORROW, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA,BERKELEY, 749 EVANS HALL,BERKELEY,CA 94720 Email address: [email protected]

ALEXANDRU ZAHARESCU,DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN, UR- BANA, ILLINOIS, UNITED STATES AND INSTITUTE OF MATHEMATICSOFTHE ROMANIAN ACADEMY,BUCHAREST, RO-70700, ROMANIA Email address: [email protected]