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Ghosts and congruences for $p^s$-approximations of hypergeometric periods

Alexander Varchenko, Wadim Zudilin

Abstract

We prove general Dwork-type congruences for constant terms 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 hypergeometric and KZ equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the simplest example of a $p$-adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of the monodromy group.

Ghosts and congruences for $p^s$-approximations of hypergeometric periods

Abstract

We prove general Dwork-type congruences for constant terms 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 hypergeometric and KZ equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the simplest example of a -adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of the monodromy group.

Paper Structure

This paper contains 38 sections, 33 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. On ghosts
  3. Mellit's theorem
  4. Convex polytopes
  5. Definition of ghosts
  6. Composed ghosts
  7. Indecomposable tuples
  8. Polynomials $I_\lambda$
  9. Admissible tuples of Laurent polynomials
  10. Dwork congruence for tuples of Laurent polynomials
  11. $p^s$-Approximation of a hypergeometric integral
  12. Function $_2F_1(\frac{1}{2},\frac{1}{2}; 1,x)$
  13. Polynomials $P_s(x)$
  14. Baby congruences
  15. Congruences for $P_s(x)$
  16. ...and 23 more sections

Key Result

Theorem 2.1

Let $a,b,c$ be tuples of integers from $\{1,\dots,p-1\}$, where $b,c,a'$ can be empty, that is, of length 0. Then

Theorems & Definitions (69)

  • Theorem 2.1: Anton Mellit, 2009, Me, unpublished
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • ...and 59 more