Local acyclicity in $p$-adic cohomology

p-adic Cohomology

The purpose of this paper is to survey some recent results in the theory of “p-adic cohomology”, by which we will mean several different (but related) things: the de Rham or p-adic étale cohomology of varieties over p-adic fields, or the rigid cohomology of varieties over fields of characteristic p > 0. Our goal is to update Illusie’s beautiful 1994 survey [I] by reporting on some of the many i...

Eisenstein Cohomology and p - adic L - Functions

§0. Introduction. In this paper, we define the module D̃(V ) of distributions with rational poles on a finite dimensional rational vector space a V . This is an infinite dimensional vector space over Q endowed with a natural action of the reductive group GV := Aut(V ). Indeed, this action extends to a natural action of the adelic group GV (AQ). For each prime p, we define we define the notion of...

Rigid cohomology and p - adic point counting ∗

I consider the problem of computing the zeta function of an algebraic variety defined over a finite field. This problem has been pushed into the limelight in recent years because of its importance in cryptography, at least in the case of curves. Wan’s excellent survey article gives an overview of what has been achieved, and what remains to be done, on the topic [15]. The purpose of this exposit...

Computing Zeta Functions via p-Adic Cohomology

We survey some recent applications of p-adic cohomology to machine computation of zeta functions of algebraic varieties over finite fields of small characteristic, and suggest some new avenues for further exploration.

p-Adic deformations of arithmetic cohomology