Number Theory Seminar

Let K be a p-adic field with absolute Galois group G_K. Integral p-adic Hodge theory gives a comparison functor T_\star from a category of linear algebra objects constructed by Breuil to the category of G_K-stable lattices inside crystalline representations with Hodge-Tate weights in [0,r]. Both these categories are equipped with a notion of short exact sequence, but they fail to be abelian. This causes certain pathologies that don't appear in the classical theory of p-adic Galois representations. Namely, one can show that the comparison functor T_\star is an exact equivalence, but its quasi-inverse is not exact. In this talk I will present recent joint work with Pavel Coupek and Adriano Marmora, where we prove that the quasi-inverse of T_\star preserves short exact sequences of a very special kind, namely those that correspond to 1-extensions of lattices inside crystalline representations with Hodge-Tate weights in [0,r] in the sense of Bloch and Kato. As a consequence, we deduce that the finite integral Bloch-Kato Selmer group H^1_f(K,T) of a crystalline lattice T can be computed by a group of 1-extensions of Breuil modules.  I will also discuss applications to 2-extensions, namely to the cup product map obtained by the Kummer sequences of two abelian varieties with good reduction. This result has potential applications to algebraic cycles.