$A$-hypergeometric series and a $p$-adic refinement of the Hasse-Witt matrix

Abstract

We identify the $p$-adic unit roots of the zeta function of a projective hypersurface over a finite field of characteristic $p$ as the eigenvalues of a product of special values of a certain matrix of $p$-adic series. That matrix is a product $F({\Lambda }^p{)}^{-1}F(\Lambda )$, where the entries in the matrix $F(\Lambda )$ are $A$-hypergeometric series with integral coefficients and $F(\Lambda )$ is independent of $p$.

28 pages