Titre de la dissertation: Wild potential good reduction of low dimensional Abelian varieties

Promoteur de thèse: Madame Maja Volkov

Résumé de la dissertation:  Let p be a prime number. Let K be a finite extension of Qp, the field of p-adic numbers, and let A be an Abelian variety over K. To such a geometric object is attached a p-adic representation of the absolute Galois group G_K of K, namely its p-adic Tate module Vp(A). These varieties together with their representations have now become classical objects in Arithmetic Geometry, and have been extensively studied since the 1960’s. However there still are quite a few open questions in the subject.

The Abelian variety A/K has good reduction if the special fibre of its Néron model remains an Abelian variety. Let L/K be a finite extension and let A_L be the extension of scalars of A to L. We say that A has potential good reduction over L if A_L has good reduction. This geometric property is carried over the associated representation Vp(A),  which is then potentially crystalline. Thanks to J.-M. Fontaine’s p-adic Hodge Theory, such a representation is described by its associated filtered (phi,G)-module, a purely semi-linear object. Now the extension L/K can either be unramified, tamely ramified, or wildly ramified. As the terminology suggests, the latter is the worst case scenario.

In 2005, Maja Volkov ([Vo05]) gave a characterization of the p-adic representations V of G_Qp arising from Abelian varieties over Qp with tame potential good reduction. This characterization is given in terms of necessary and sufficient conditions on the filtered (phi,G)-module associated to V. In this thesis we are interested in Abelian varieties defined over K with wild potential good reduction, that is, Abelian varieties that acquire good reduction over a wildly ramified extension. Not much is known in this situation, and our purpose is to provide the first explicit examples in low dimension.

A well-known result of Serre and Tate states that when p>2dim(A)+1 the potential good reduction is necessarily tame. For this reason we concentrate on elliptic curves over Q3 (i.e. dim(A)=1 and p=3) and Abelian surfaces over Q3 and Q5 (i.e. dim(A)=2 and p=3 or 5).

In the first part of this thesis we provide a full classification of the 3-adic representations of G_Q3 arising from elliptic curves over Q_3 with potential good reduction. Our classification is obtained as follows. We begin by computing the filtered (phi,G)-modules satisfying the conditions from [Vo05]. Then we show that each representation obtained in this way indeed arises from some elliptic curve over Q3, which is achieved by constructing minimal Galois pairs introduced in [Vo05]. This classification highlights new phenomena specific to the wild potential good reduction case.

The second part of this thesis is dedicated to Abelian surfaces. Here we do not aim to classify their attached representations, but merely to investigate the inertia subgroups

naturally appearing in a situation of potential good reduction, with a focus on the wild ones. In 2005 Alice Silverberg and Yuri Zarhin have classified all such possible inertia subgroups over a discretely valued field with perfect residue field. However most of the wild ones are achieved over local fields in equicharacteristic. We show that each such group can actually be realised in mixed characteristic. This result is obtained in three steps. We begin by realising each of these groups as the inertia subgroup of a finite extension of Qp. We then provide a minimal polarised Galois pair for each such extension. We next use a recent result of Séverin Philip that guarantees the existence of appropriate filtrations on their (phi,G)-modules, thus allowing us to lift our Galois pairs to characteristic zero and achieving our goal.