2017/01/17 by Alan Adolphson, Steven Sperber, Adolphson, Alan +1
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.1701.04509
18 pages
arxiv created 2017/01/17 · arxiv updated 2017/01/18
The Hasse-Witt matrix of a hypersurface in \mathbb Pn 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 ≤ 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.