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GHOSTS AND CONGRUENCES FOR $\boldsymbol {p}^{\boldsymbol {s}}$-APPROXIMATIONS OF HYPERGEOMETRIC PERIODS

Part of: Diophantine equations Other special functions Deformations of analytic structures Differential and difference algebra

Published online by Cambridge University Press:  02 August 2023

ALEXANDER VARCHENKO  and
WADIM ZUDILIN Open the ORCID record for WADIM ZUDILIN [Opens in a new window]
ALEXANDER VARCHENKO*
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
WADIM ZUDILIN
Affiliation:
Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, PO Box 9010, 6500 GL Nijmegen, The Netherlands e-mail: w.zudilin@math.ru.nl
*
e-mail: anv@email.unc.edu
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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 Knizhnik–Zamolodchikov (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 its monodromy representation.

Keywords

hypergeometric equationKZ equationsDwork congruencesmaster polynomialsps-approximation polynomials

MSC classification

Primary: 11D79: Congruences in many variables
Secondary: 12H25: $p$-adic differential equations 32G34: Moduli and deformations for ordinary differential equations (e.g. Knizhnik-Zamolodchikov equation) 33C05: Classical hypergeometric functions, $_2F_1$ 33E30: Other functions coming from differential, difference and integral equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 116 , Issue 1 , February 2024 , pp. 96 - 127
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Michael Coons

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