Applications of Generalized Fermat Varieties to Zeta Functions of Artin-Schreier Curves.

[摘要] We outline an approach to studying Artin-Schreier curves X_f (associated with equations of the form y^q-y=f(x)) involving auxiliary varieties of higher dimension. Specifically, for a fixed f(x) and every positive integer n we consider the affine variety defined by the equation sum_{i=1}^n f(x_i) and its projective closure. S_n acts on these varieties and thus on their cohomology groups. We explain what these S_n-representations ;;know;; and relate properties of their irreducible decompositions to arithmetic properties of X_f. This relationship has a number of applications, and in particular we explain how to deduce bounds for the number of F_{q^n} rational-points on X_f as well as the existence of certain algebraic relations among the zeroes of the zeta function of X_f.

[发布机构] University of Michigan

[效力级别] Algebraic Geometry

[关键词] Number Theory;Algebraic Geometry;Exponential Sums;Artin-Schreier Curve;Mathematics;Science;Mathematics