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

Abstract

Let V be a p-adic representation of the absolute Galois group G of Qp that becomes crystalline over a finite tame extension, and assume p = 2. We provide necessary and sufficient conditions for V to be isomorphic to the p-adic Tate module Vp(A) of an abelian variety A defined over Qp. These conditions are stated on the filtered (ϕ, G)module attached to V .

On descending cohomology geometrically

“…In a similar spirit (but necessarily more involved fashion), Volkov [Vol05] has characterized the potentially crystalline -adic representations of which arise as summands of the -adic Tate module of an abelian variety over . As with finite fields, this lets one conclude that if is a smooth projective threefold with , and if acquires good reduction over a tamely ramified extension of , then is isomorphic to a subrepresentation of for some abelian variety .…”

Abelian Surfaces With Supersingular Good Reduction and Non-Semisimple Tate Module

“…The existence of such surfaces follows from the characterization of p-adic representations of G arising from abelian varieties with (tame) potential good reduction obtained in [10], and indeed provides an example of application of this result. In order to explicitly describe our objects, we use Fontaine's contravariant functor estab-lishing an equivalence between crystalline p-adic representations of G and admissible filtered ϕ-modules.…”

“…In Sec. 1, we briefly review this theory as well as the characterization in [10,Theorem 1.2], and outline the general strategy. In Secs.…”

“…The proof of Theorem 5.7 in[10] details this construction.Thus we want to construct an admissible filtered ϕ-module (D, Fil) over Q p satisfying conditions (1)-(3) of Theorem 1.2 and such that (a) P char (ϕ) is a supersingular p-Weil polynomial (b) (D, Fil) is not semisimple.…”

Hodge theory of $p$-adic varieties: a survey

“…The last inclusion in (2.28) follows from Theorem 2.4. This inclusion is strict: we did not put any restrictions on eigenvalues of Frobenius (we note however that these restrictions could be added, see [78] for an example) but, as shown in [77], even adding such restrictions would not have been sufficient to force the equality.…”