Comparisons of cohomology theories of (phi,Gamma)-modules

par Rustam Steingart

jeudi 20 févr. 2025, 14:00 → 16:00 Europe/Paris

The category of L-linear representations of the absolute Galois group G_L of a finite extension L of Q_p is equivalent to the category of étale (phi,Gamma)-modules.

If L\neq Q_p, there are representations which fail to be overconvergent. A theorem of Laurent Berger asserts that so-called analytic representations are overconvergent.

I will introduce "overconvergent" and "analytic" cohomology. The H^1's are subgroups of Galois cohomology and parametrise extensions within the above subcategories.

I will explain how to pass between these cohomology theories, how (and why) they (do not) compare with Galois cohomology and suggest a different view point motivated by Dolbeaut's Lemma in complex analysis.