Meromorphic continuation for the zeta function of a Dwork hypersurface (Q625346)

From the Introduction: ``In [Ann. Math. (2) 171, No. 2, 779--813 (2010; Zbl 1263.11061)] \textit{M. Harris, N. Shepherd-Barron} and \textit{R. Taylor} have proved a potential modularity theorem, showing that certain Galois representations become automorphic after a sufficiently large totally real base change. In their argument, a key role is played by certain families of hypersurfaces, called Dwork families -- in particular, by the part of the cohomology of the family which is invariant under a certain group action (We will write \(\mathcal{F}\) for the motive given by this part of the cohomology).'' This invariant part is proven to have a controlled behaviour so that, in particular, Harris and others can prove the meromorphic continuation and functional equation for the zeta function of \(\mathcal{F}\). The aim of the paper under review is to study whether the same control can be gained on the whole cohomology of the Dwork family. Recall that the Dwork family is defined to be the \(\mathbb{Z}[1/N,\mu_N]\)-scheme \(Y\subseteq \mathbb{P}^{N-1}\times\mathbb{P}^1\) defined by the equations \[ \mu(X_1^N+\cdots+X_N^N)=N\lambda X_1X_2\cdots X_N, \] using \((\mu,\lambda)\) and \((X_1:\cdots:X_N)\) as coordinates in \(\mathbb{P}^1\) and \(\mathbb{P}^N\), respectively. The author proves that if \(N=5\), the zeta function of the cohomology of \(Y\) admits a meromorphic continuation. This is done by studying the part of the cohomology which is not considered in [loc. cit.] through a very explicit calculation of the \(S_5\)-action on it (Section 3). Section 4 hosts the proof of the main theorem (Theorem 7 and Corollary 8) while Section 5 contains some final remark.