Monomial Deformations of Certain Hypersurfaces and Two Hypergeometric Functions

TL;DR

This paper explicitly describes the zeta functions of certain hypersurfaces using hypergeometric functions over finite fields, combining character sums and weight theory to facilitate point counting and zeta function calculation.

Contribution

It provides a new explicit description of zeta functions for a class of hypersurfaces via hypergeometric functions, extending Dwork family analysis.

Findings

Explicit formulas for zeta functions in terms of hypergeometric functions

Method combining character sums and weight theory for point counting

Generalization of Dwork family hypersurfaces

Abstract

The purpose of this article is to give an explicit description, in terms of hypergeometric functions over finite fields, of zeta function of a certain type of smooth hypersurfaces that generalizes Dwork family. The point here is that we count the number of rational points employing both character sums and the theory of weights, which enables us to enlighten the calculation of the zeta function.