Hypergeometric Functions over Finite Fields and their relations to Algebraic Curves

TL;DR

This paper establishes a connection between hypergeometric functions over finite fields, the point counts on algebraic curves, and the roots of their zeta functions, proposing a conjecture linking these concepts.

Contribution

It introduces explicit relations between hypergeometric functions and algebraic curve point counts, and relates these functions to zeta function roots, advancing understanding in finite field arithmetic.

Findings

Explicit formulas linking hypergeometric functions and point counts

Relations between hypergeometric functions and zeta function roots

A conjecture connecting these mathematical objects

Abstract

In this work we present an explicit relation between the number of points on a family of algebraic curves over $\F_{q}$ and sums of values of certain hypergeometric functions over $\F_{q}$. Moreover, we show that these hypergeometric functions can be explicitly related to the roots of the zeta function of the curve over $\F_{q}$ in some particular cases. A general conjecture relating these last two is presented and advances toward its proof are shown in the last section.