Three vignettes on hypergeometric normal functions

TL;DR

This paper employs Hodge-theoretic techniques to elucidate number-theoretic identities, analyze Abel-Jacobi period functions, and prove Golyshev's conjecture on algebraic hypergeometric functions.

Contribution

It introduces novel Hodge-theoretic methods to connect hypergeometric functions with number theory and algebraic geometry, providing new proofs and insights.

Findings

Explanation of identities related to hypergeometric functions

Description of Frobenius duals of Abel-Jacobi periods

A new proof of Golyshev's conjecture

Abstract

We use Hodge-theoretic methods to (i) explain number-theoretic identities of a type recently considered by Guillera and Zudilin, (ii) describe the Frobenius dual of Abel-Jacobi period functions, and (iii) offer a new proof of Golyshev's conjecture on algebraic hypergeometrics aided by an argument in the spirit of Lefschetz's (1,1) theorem.