2021
DOI: 10.4171/dm/843
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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 u… Show more

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Cited by 2 publications
(1 citation statement)
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“…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].…”
Section: Introductionmentioning
confidence: 99%
“…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].…”
Section: Introductionmentioning
confidence: 99%