The $\ell$-adic hypergeometric function and associators

TL;DR

This paper introduces an $-adic analogue of Gauss's hypergeometric function linked to Galois actions, establishing foundational properties and connecting it to associators and gamma functions.

Contribution

It defines a novel $$-adic hypergeometric function inspired by complex hypergeometric equations and explores its fundamental properties and connections to associators.

Findings

Established $$-adic hypergeometric function properties analogous to classical identities.

Connected the $$-adic function to associators and gamma functions.

Extended previous results by Ohno and Zagier.

Abstract

We introduce an $\ell$-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 $\ell$-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.