Meromorphic continuation for the zeta function of a Dwork hypersurface

Barnet-Lamb, Thomas

doi:10.2140/ant.2010.4.839

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Meromorphic continuation for the zeta function of a Dwork hypersurface

Thomas Barnet-Lamb

Vol. 4 (2010), No. 7, 839–854

DOI: 10.2140/ant.2010.4.839

Abstract

We consider the one-parameter family of hypersurfaces in $ℙ^{5}$ over $ℚ$ with projective equation $(X_{1}^{5} + X_{2}^{5} + X_{3}^{5} + X_{4}^{5} + X_{5}^{5}) = 5 t X_{1} X_{2} \dots X_{5}$ , 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.

Keywords

Dwork hypersurface, potential automorphy, zeta function

Mathematical Subject Classification 2000

Primary: 11G40

Secondary: 11R39, 11F23

Milestones

Received: 16 January 2009

Revised: 9 August 2010

Accepted: 10 August 2010

Published: 29 January 2011

Authors

Thomas Barnet-Lamb
Department of Mathematics Brandeis University 415 South Street MS 050 Waltham, MA 02453 United States
http://people.brandeis.edu/~tbl/

Vol. 4, No. 7, 2010