A report on the cohomological coprimality of Galois representations (Algebraic Number Theory and Related Topics 2014)

Abstract

Let K be a local or a global field and GK its absolute Galois group. In this note we announce some results on the vanishing of certain Galois cohomology groups associated with representations of GK coming from geometry. Motivated by our efforts to generalize some results of Coates, Sujatha and Wintenberger, we introduce the notion of "cohomological co-primality", which provides another notion of independence between such representations. We consider proper smooth varieties X and X' over a p-adic field K with potential good reduction. Then it can be shown that in many cases where X and X' have "quite different" nature, their corresponding representations are cohomologically coprime. When K is a number field, we can prove the cohomological coprimality of systems of ℓ-adic representations of GK associated with elliptic curves which are non-isogenous over K.