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

Alexander Varchenko, Wadim Zudilin

Abstract

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, 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 $y^2=(t-z_1)\dots (t-z_{2g+1})$ 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.

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

Abstract

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

Paper Structure

This paper contains 25 sections, 31 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. On ghosts
  3. Definition of ghosts
  4. Admissible tuples
  5. Hasse--Witt matrix
  6. Derivatives
  7. Convergence
  8. Infinite tuples
  9. Domain of convergence
  10. KZ equations
  11. KZ equations
  12. Solutions over ${\mathbb C}$
  13. Solutions as vectors of first derivatives
  14. Solutions modulo $p^s$
  15. Congruences for solutions of KZ equations
  16. ...and 10 more sections

Key Result

Lemma 2.1

For $s=0,1,\dots,l$, we have $V_s(x) \equiv 0 \pmod{p^s}$.

Theorems & Definitions (61)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Example
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • ...and 51 more