An algebro-geometric study of special values of hypergeometric functions ${}_3F_2$

TL;DR

This paper investigates conditions under which special values of certain hypergeometric functions can be expressed using logarithms of algebraic numbers, employing algebro-geometric methods related to higher regulators.

Contribution

It provides new sufficient conditions and two algebro-geometric proofs connecting hypergeometric function values to algebraic logarithms.

Findings

Identifies conditions for hypergeometric values to be expressed via algebraic logarithms

Develops two independent algebro-geometric proofs

Links hypergeometric values to higher regulators

Abstract

For certain class of hypergeometric functions ${}_3F_2$ with rational parameters, we give a sufficient condition for the special value at $1$ to be expressed in terms of logarithms of algebraic numbers. We give two proofs, both of which are algebro-geometric and related to higher regulators.