Congruences for Hasse-Witt matrices and solutions of $p$-adic KZ equations

We prove general Dwork-type congruences for Hasse–Witt matrices attached to tuples of Laurent polynomials.We apply this result to establishing arithmetic and p-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of Knizhnik–Zamolodchikov (KZ) equations, the solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the $p$-adic KZ connection associated with the family of hyperelliptic curves $y^2 = (t - z_1) \dotsc (t - z_{2g+1})$ has an invariant subbundle of rank $g$. Notice that the corresponding complex KZ connection has no nontrivial subbundles due to the irreducibility of its monodromy representation.

Keywords

KZ equations, Dwork congruences, master polynomials, Hasse–Witt matrices

2010 Mathematics Subject Classification

Primary 11D79. Secondary 12H25, 32G34, 33C05, 33E30.

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The first-named author was supported in part by NSF grant DMS-1954266.

Received 25 October 2021

Accepted 24 December 2022

Published 26 March 2024