Cohomology and Overconvergence for Representations of Powers of Galois Groups

Abstract

We show that the Galois cohomology groups of p-adic representations of a direct power of Gal(Q p /Q p ) can be computed via the generalization of Herr's complex to multivariable (ϕ, Γ)-modules. Using Tate duality and a pairing for multivariable (ϕ, Γ)modules we extend this to analogues of the Iwasawa cohomology. We show that all padic representations of a direct power of Gal(Q p /Q p ) are overconvergent and, moreover, passing to overconvergent multivariable (ϕ, Γ)-modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.

Multivariable de Rham representations, Sen theory and $p$-adic differential equations

“…This is the analogue to that introduced by Fontaine in [16], and used by Berger in [5]. In particular, we show that a p-adic representation V of G K,∆ is de Rham if and only if the associated module with connection D dif (V ) is trivial (proposition 5.18), and relate our construction to that of overconvergent (ϕ, Γ)-modules developed by Pal-Zábrádi (cf [20]) and Carter-Kedlaya-Zábrádi (cf [12]) by an analogue of [5,Corollaire 5.8] (cf theorem 5.23).…”

“…In fact the geometric base for our objects is merely a finite discrete space (cf remark 3.23). Secondly, this article should be seen as a first step towards the introduction of the multivariable periods rings B cris and B st , and eventually, a step in the direction of full analogue of Berger's results via the theory of p-adic differential systems in several variables (as was again foreseen in [20]).…”

“…in terms of period rings and (ϕ, Γ)-modules. This second approach has been pursued in recent works by Zábrádi,Pal,Kedlaya and Carter ([26], [27], [20] and [12]) in terms of multivariable (multivariate) (ϕ, Γ)-modules associated to p-adic representations of G K,∆ .…”

“…has a full set of solutions) if and only if the G K,∆ -representation V is de Rham. Moreover, we relate the differential module D dif (V ) with overconvergent (ϕ, Γ)-module arising from Pal-Zábrádi theory (cf [20]).…”

“…We prove that this system is trivial if and only if the representation V is de Rham. Finally, we relate this differential system to the multivariable overconvergent (ϕ, Γ)-module of V constructed by Pal and Zábrádi in [20], along classical Berger's construction [5].…”