Isotypic Decomposition of the Cohomology and Factorization of the Zeta Functions of Dwork Hypersurfaces

TL;DR

This paper demonstrates how analyzing the automorphism group representations in etale cohomology of Dwork hypersurfaces enables the factorization of their zeta functions, linking geometric symmetries to arithmetic properties.

Contribution

It provides a concrete example of using group representation theory in etale cohomology to factor zeta functions of hypersurfaces, illustrating a method for connecting geometry and number theory.

Findings

Factorization of zeta functions via automorphism group representations

Application of etale cohomology to Dwork hypersurfaces

Illustration of the link between symmetries and zeta function properties

Abstract

The aim of this article is to illustrate, on the example of Dwork hypersurfaces, how the study of the representation of a finite group of automorphisms of a hypersurface in its etale cohomology allows to factor its zeta function.