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The $\ell$-adic hypergeometric function and associators

Hidekazu Furusho

Vol. 5 (2023), No. 1, 1–29
Abstract

We introduce an -adic analogue of Gauss’s hypergeometric function arising from the Galois action on the fundamental torsor of the projective line minus three points. Its definition is motivated by a relation between the KZ equation and the hypergeometric differential equation in the complex case. We show two basic properties, analogues of Gauss’s hypergeometric theorem and of Euler’s transformation formula for our -adic function. We prove them by detecting a connection of a certain two-by-two matrix specialization of even unitary associators with the associated gamma function, which extends the result of Ohno and Zagier.

Keywords
associators, KZ equation, hypergeometric functions.
Mathematical Subject Classification
Primary: 11F80
Secondary: 11M32, 33C05
Milestones
Received: 23 October 2021
Revised: 31 July 2022
Accepted: 29 August 2022
Published: 20 April 2023
Authors
Hidekazu Furusho
Graduate School of Mathematics
Nagoya University
Japan