Wojciech J. Gajda

Abstract:

For a hypersurface X over a ring R we construct a sequence of matrices with coefficients in R whose reductions modulo p describe iterates of the Hasse--Witt operation (whenever the latter is defined). We show that our matrices satisfy a reach system of congruences modulo powers of p. If the Hasse-Witt operator is invertible these congruences yield p-adic limit formulas, which conjecturally describe the Gauss-Manin connection and the Frobenius operator on the unit-root F-crystal constructed in the 80s by Nicholas Katz.