Meromorphic continuation for the zeta function of a Dwork hypersurface

Abstract

We consider the one-parameter family of hypersurfaces in ${\mathbb{P}}^5$ over $\mathbb{Q}$ with projective equation $\left(X_1^5+X_2^5+X_3^5+X_4^5+X_5^5\right)=5t{X}_1{X}_2\dots X_5$, proving that the Galois representations attached to their cohomologies are potentially automorphic, and hence that the zeta function of the family has meromorphic continuation to the whole complex plane.