Vol. 4, No. 7, 2010

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Meromorphic continuation for the zeta function of a Dwork hypersurface

Thomas Barnet-Lamb

Vol. 4 (2010), No. 7, 839–854
Abstract

We consider the one-parameter family of hypersurfaces in ${ℙ}^{5}$ over $ℚ$ 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.

Keywords
Dwork hypersurface, potential automorphy, zeta function
Mathematical Subject Classification 2000
Primary: 11G40
Secondary: 11R39, 11F23
Milestones
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/