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 expository article is to extract the essential content of previous results, and contrast this with some new developments in p-adic point counting. All of the p-adic algorithms I discuss rely upon rigid cohomology in some incarnation, and the alternative approaches pioneered by Mestre and Satoh are not touched upon. Moreover, little attention is paid to the precise running times of algorithms, the focus instead being on the qualitative nature of the complexities of algorithms. For this reason, much significant recent work using rigid cohomology, by Denef, Gaudry, Gerkmann, Gürel, Vercauteren and others, is not mentioned. Let Fq be a finite field with q elements of characteristic p. Let X be an algebraic variety defined over Fq. For each positive integer k, denote by Nk the number of Fqk -rational points on X. The zeta function Z(X, T ) of X is the formal power series