Drinfeld's lemma for perfectoid spaces and overconvergence of multivariate $(\varphi, \Gamma)$-modules

Abstract

Let p be a prime, let K be a finite extension of Q p , and let n be a positive integer. We construct equivalences of categories between continuous p-adic representations of the n-fold product of the absolute Galois group G K and (ϕ, Γ)-modules over one of several rings of n-variable power series. The case n = 1 recovers the original construction of Fontaine and the subsequent refinement by Cherbonnier-Colmez; for general n, the case K = Q p had been previously treated by the third author. To handle general K uniformly, we use a form of Drinfeld's lemma on the profinite fundamental groups of products of spaces in characteristic p, but for perfectoid spaces instead of schemes. We also construct the multivariate analogue of the Herr complex to compute Galois cohomology; the case K = Q p had been previously treated by Pal and the third author, and we reduce to this case using a form of Shapiro's lemma. 2020 Mathematics Subject Classification: 11S37 Keywords and Phrases: (ϕ, Γ)-modules, perfectoid spaces 2 Notation Fix a finite extension K/Q p , an algebraic closure K alg of K, and a finite set ∆ with n elements. We are interested in continuous Z p -representations of the group G K,∆ := α∈∆ Gal(K alg /K).Along the way, we will also encounter the groupsAs mentioned in the introduction, one can also handle products of Galois groups of possibly distinct finite extensions of Q p , but to simplify notation we postpone discussion of this case until Section 8.

Drinfeld's lemma for F-isocrystals, II: Tannakian approach

“…Inspired by Berger's thesis [Ber02], we expect to deduce a ‘de Rham implies potentially semistable’ result for -adic representations of powers of Galois groups from local Drinfeld's lemma. There are some related results in this direction: the overconvergence of multivariate -modules has been proved by the first author, Carter, and Zábrádi [CKZ21]; multivariable de Rham representations and the associated -adic differential equations are studied by Brinon, Chiarellotto, and Mazzari [BCM21].…”