A class of p-adic Galois representations arising from abelian varierties over Q_p

We give a complete classification of all the potentially crystalline 3-adic representations of the absolute Galois group of $\mathbb{Q}_3$ that are isomorphic to the Tate module of an elliptic curve defined over $\mathbb{Q}_3$. These representations are described in terms of their associated filtered $(\varphi, \mathrm{Gal}(K/\mathbb{Q}_3))$-modules. The most interesting cases occur when the potential good reduction is wild.

This paper is devoted to the study of the $\ell$-adic representations of the absolute Galois group $G$ of ${\mathbb Q}_p$, $p\geq 5$, associated to an elliptic curve over ${\mathbb Q}_p$, as $\ell$ runs through the set of all prime numbers (including $\ell =p$, in which case we use the theory of potentially semi-stable $p$-adic representations). For each prime $\ell$, we give the complete list of isomorphism classes of ${\mathbb Q}_{\ell}[G]$-modules coming from an elliptic c...

Let k be a perfect field of characteristic p>0. When p>2, Fontaine and Laffaille have classified p-divisibles groups and finite flat p-groups over the Witt vectors W(k) in terms of filtered modules. Still assuming p>2, we extend these classifications over an arbitrary complete discrete valuation ring A with unequal characteristic (0,p) and residue field k by using "generalized" filtered modules. In particular, there is no restriction on the ramification index. In the case k i...

Let $K$ be a mixed characteristic complete discrete valuation field with perfect residue field of characteristic $p$. We construct a new linear category called the category of crystalline $(\varphi,\Gamma)$-modules over $\widetilde{\mathbb{A}}_K^{+}$ and show that it is equivalent to the category of crystalline $\mathbb{Z}_p$-representations of the absolute Galois group of $K$. In other words, we determine the $(\varphi,\Gamma)$-modules over $\widetilde{\mathbb{A}}_K$ which c...

For a finite extension $K/\mathbb{Q}_p$ and a split reductive group $G$ over $\mathcal{O}_K$, let $\overline{\rho} \colon \mathrm{Gal}_K \to G(\overline{\mathbb{F}}_p)$ be a continuous quasi-semisimple mod $p$ $G$-valued representation of the absolute Galois group $\mathrm{Gal}_K$. Let $\overline{\rho}^{\mathrm{ab}}$ be the abelianization of $\overline{\rho}$ and fix a crystalline lift $\psi$ of $\overline{\rho}^{\mathrm{ab}}$. We show the existence of a crystalline lift $\rh...

Let V be a crystalline p-adic representation of the absolute Galois group G_K of an finite unramified extension K of Q_p and T a lattice of V stable by G_K. We prove the following result: Let Fil^1 V be the maximal sub-representation of V with Hodge-Tate weights strictly positive and Fil^1 T=T \cap Fil^1 V. Then, the projective limit of the H^1_g(K(\mu_{p^n}), T) is equal up to torsion to the projective limit of the H^1(K(\mu_{p^n}), Fil^1 T). So its rank over the Iwasawa a...

We study Galois representations attached to nonsimple abelian varieties over finitely generated fields of arbitrary characteristic. We give sufficient conditions for such representations to decompose as a product, and apply them to prove arithmetical analogues of results shown by Moonen and Zarhin in the context of complex abelian varieties (of dimension at most 5).

We propose the notion of the {\em crystalline sub-representation functor} defined on $p$-adic representations of the Galois groups of finite extensions of $\Qp$, with certain restrictions in the case of integral representations. By studying its right-derived functors, we find a natural extension of a formula of Grothendieck expressing the group of connected components of a Neron model of an abelian variety in terms of Galois cohomology.

Let K be a number field and {V_l} be a rational strictly compatible system of semisimple Galois representations of K arising from geometry. Let G_l and V_l^ab be respectively the algebraic monodromy group and the maximal abelian subrepresentation of V_l for all l. We prove that the system {V_l^ab} is also a rational strictly compatible system under some group theoretic conditions, e.g., when G_l' is connected and satisfies Hypothesis A for some prime l'. As an application, we...

In this paper we develop a theory of class invariants associated to $p$-adic representations of absolute Galois groups of number fields. Our main tool for doing this involves a new way of describing certain Selmer groups attached to $p$-adic representations in terms of resolvends associated to torsors of finite group schemes.