Meromorphic continuation for the zeta function of a Dwork hypersurface

TL;DR

This paper proves that the zeta function of a family of Dwork hypersurfaces in projective 5-space has a meromorphic continuation across the entire complex plane by establishing the potential automorphy of associated Galois representations.

Contribution

It demonstrates the potential automorphy of Galois representations linked to these hypersurfaces, leading to meromorphic continuation of their zeta functions.

Findings

Zeta function admits meromorphic continuation over the complex plane.

Galois representations attached to the hypersurfaces are potentially automorphic.

Establishes a link between hypersurface cohomology and automorphic forms.

Abstract

We consider the one-parameter family of hypersurfaces in $\Pj^5$ with projective equation (X_1^5+X_2^5+X_3^5+X_4^5+X_5^5) = 5\lambda X_1 X_2... X_5, (writing $\lambda$ for the parameter), proving that the Galois representations attached to their cohomologies are potentially automorphic, and hence that the zeta function of the family has meromorphic continuation throughout the complex plane.