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arXiv · 1701.04509

A generalization of the Hasse-Witt matrix of a hypersurface

Alan Adolphson·Steven Sperber
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Abstract

The Hasse-Witt matrix of a hypersurface in ${\mathbb P}^n$ over a finite field of characteristic $p$ gives essentially complete mod $p$ information about the zeta function of the hypersurface. But if the degree $d$ of the hypersurface is $\leq n$, the zeta function is trivial mod $p$ and the Hasse-Witt matrix is zero-by-zero. We generalize a classical formula for the Hasse-Witt matrix to obtain a matrix that gives a nontrivial congruence for the zeta function for all $d$. We also describe the differential equations satisfied by this matrix and prove that it is generically invertible.

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Alan Adolphson, Steven Sperber. 2017-01-17. A generalization of the Hasse-Witt matrix of a hypersurface. https://arxiv.org/abs/1701.04509

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