ON p-ADIC UNIFORMIZATION OF ABELIAN VARIETIES WITH GOOD REDUCTION ADRIAN IOVITA, JACKSON S. MORROW, AND ALEXANDRU ZAHARESCU, WITH AN APPENDIX BY YEUK HAY JOSHUA LAM AND ALEXANDER PETROV ABSTRACT. Let p be a rational prime, let F denote a finite, unramified extension of Qp, K the maximal unramified extension of Qp, K some fixed algebraic closure of K, and Cp the completion of K. Let GF the absolute Galois group of F. Let A be an abelian variety defined over F, with good reduction. Classically, the Fontaine integral was seen as a Hodge–Tate comparison morphism, i.e. as a map C → C ϕA ⊗ 1Cp : Tp(A) ⊗Zp p Lie(A)(F) ⊗F p(1), and as such it is surjective and has a large kernel. The present article starts with the observation that if we do not tensor Tp(A) with Cp, then the GK Fontaine integral is often injective. In particular, it is proved that if Tp(A) = 0, then ϕA is injec- tive. We give two applications of this result. First, we extend the Fontaine integral to a perfectoid like G universal cover of A and show that if Tp(A) K = 0 then A(K) has a type of p-adic uniformization, which resembles the classical complex uniformization. Second, we prove results concering the ram- ification properties of the field extension of K obtained by adjoining the coordinates of the p-power torsion points of the formal group of A. 1. Introduction As a consequence of a series of articles starting with Tate’s original [Tat67] and ending with the article of Scholze–Weinstein [SW13], we now have a complete classification of the p-divisible groups over OCp in terms of their Hodge–Tate structures. We believe it is a legitimate question to wonder if a similar classification exists for abelian varieties over a p-adic field. The goal of this work is to investigate this question. This work has two main inspiration sources: [Fon03] and [BK07]. To start with, let us recall that in the Introduction, in fact in the Prologue of [Fon03], J.-M. Fontaine presents the following picture. Let K denote the completion of a number field and choose an algebraic closure K of K. We denote the absolute Galois group of K by GK := Gal(K/K) and by C the completion of K. Let A denote an abelian variety over K of dimension g > 1, and let Lie(A) denote the Lie algebra of A, arXiv:2107.09165v2 [math.NT] 20 Sep 2021 seen as a vector group over K. Let us first suppose K is archimedean. In this case we have K = R or K = C, K = C = C, GK = {1, τ}, with τ the complex conjugation, or GK = {1}. The exponential map is everywhere defined and defines the exact sequence: expA 0 H1(A(C), Z) Lie(A)(C) A(C) 0. From this, we obtain the complex uniformization of A, i.e. an isomorphism of complex Lie-groups: → → → → A(C) =∼ Lie(A)(C)/Λ =∼ Cg/Λ, where Λ is the image of the lattice H1(A(C), Z). Date: September 21, 2021. 2020 Mathematics Subject Classification. 11G10 (14K20, 11G25, 14L05). Key words and phrases. p-adic Uniformization, Abelian varieties, Fontaine integration, p-adic Dynamics. 1 Now suppose that K is a finite extension of Qp for p > 2 a rational prime. In this case, we will denote Cp := C. Then the logarithm is everywhere defined and we have a natural commutative diagram with exact rows: 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 topo- logical 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.