Abstract

. In this report, we discuss the problem of computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo p m of the zeta function of a hypersurface, where p is the characteristic of the finite field. 1991 Mathematics Subject Classification: 11Y16, 11T99, 14Q15. 1. Introduction Let p be a prime number. Let F q be a finite field of q elements of characteristic p. Let X be an algebraic variety defined over F q , say an affine variety defined by the vanishing of r polynomials in n variables: f 1 (x 1 ; \Delta \Delta \Delta ; xn ) = \Delta \Delta \Delta = f r (x 1 ; \Delta \Delta \Delta ; xn ) = 0; where the polynomials f i have coefficients in F q . Let N(X) denote the number of F q -rational points on X . Problem I. Compute N(X) efficiently. In addition to its intrinsic theoretical interest, this fundamental algorithmic problem has important applications in diverse areas such as coding theory, cryptography, ...

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