Counting points onvarieties over finite fields of small characteristic

Summary

We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing the order of the group of rational points on the Jacobian of a smooth geometrically connected projective curve over a finite field of small characteristic.

The purpose of this paper is to give an elementary and self-contained proof that one may efficiently compute zeta functions of arbitrary varieties of fixed dimension over finite fields of suitably small characteristic. This is achieved via the p-adic methods developed by Dwork in his proof of the rationality of the zeta

function of a variety over a finite field [Dwork 1960; 1962]. Dwork’s theorem shows that it is in principle possible to compute the zeta function. Our main contribution is to show how Dwork’s trace formula, Bombieri’s degree bound [1978] and a semilinear reduction argument yield an efficient algorithm for doing so. That p-adic methods may be used to efficiently compute zeta functions for small characteristic was first suggested in [Wan 1999; 2008], where Wan gives a simpler algorithm for counting the number of solutions to an equation over a finite field modulo small powers of the characteristic.