On the Zeta Function of a Family of Quintics

TL;DR

This paper proves a numerical link between the zeta functions of hypergeometric curves and quintic families using Koblitz formulas and Gauss sums, without geometric interpretation.

Contribution

It provides a proof of the observed numerical relationship between zeta functions of different algebraic families, expanding understanding of their connections.

Findings

Confirmed the numerical link between zeta functions of hypergeometric curves and quintics.

Used Koblitz formulas and Gauss sums to establish the relationship.

Did not derive geometric insights from the proof.

Abstract

In this article, we give a proof of the link between the zeta function of two families of hypergeometric curves and the zeta function of a family of quintics that was observed numerically by Candelas, de la Ossa, and Rodriguez Villegas. The method we use is based on formulas of Koblitz and various Gauss sums identities; it does not give any geometric information on the link.