Period integral of open Fermat surfaces and special values of hypergeometric functions

TL;DR

This paper provides explicit formulas for special values of hypergeometric functions 3F2 at 1, connecting them to algebraic numbers and Hodge cycles on Fermat surfaces, extending previous theoretical classifications.

Contribution

It offers explicit expressions for hypergeometric 3F2 values outside exceptional cases, building on prior classifications related to Fermat surfaces and Hodge cycles.

Findings

Explicit formulas for hypergeometric 3F2 values at 1

Connection to algebraic numbers and Hodge cycles

Extension beyond exceptional characters

Abstract

In the previous paper by Asakura-Otsubo-Terasoma, we prove that the special values of the hypergeometric function 3F2 at 1 are linear combinations of logarithms of algebraic numbers and 1 over algebraic numbers, if exponents are rational numbers satisfying a certain arithmetic condition. Aoki and Shioda completely classified these sets of rational numbers satisfying this condition in connection with Hodge cycles on Fermat surfaces. In this paper, we give an explicit expression of special values of hypergoemetricy 3F2 which does not belong to exceptional characters.