Finite hypergeometric functions

TL;DR

This paper explores finite hypergeometric functions over finite fields, linking their values to Frobenius traces and point counts on varieties, with a focus on one-variable cases with rational monodromy.

Contribution

It provides a detailed analysis of one-variable finite hypergeometric functions with rational monodromy, extending the understanding of their algebraic and geometric properties.

Findings

Values relate to Frobenius traces on l-adic sheafs

Formulas for point counts on certain varieties are derived

Focus on functions with monodromy over the rational integers

Abstract

Finite hypergeometric functions are complex valued functions on finite fields which are the analogue of the classical analytic hypergeometric functions. From the work of N.M.Katz it follows that their values are traces of Frobenius on certain l-adic sheafs. More concretely, in many instances their values can be used to give formulas for pointcounts of F_q-rational points on certain varieties. In this paper we work out the case of one-variable functions whose monodromy in the analytic case can be defined over the rational integers.