A Whipple $_7F_6$ formula revisited

TL;DR

This paper revisits Whipple's $_7F_6$ hypergeometric formula through the lens of hypergeometric data, revealing structural properties, Galois representations, and connections to automorphic forms and modular periods.

Contribution

It provides a new perspective on Whipple's formula by linking it to hypergeometric data, Galois representations, and automorphic forms, especially over $\,\mathbb{Q}$.

Findings

Hypergeometric data $HD$ are primitive and self-dual.

Galois representations associated to $HD$ are decomposable and automorphic.

Hypergeometric values $_7F_6(1)$ relate to modular form periods.

Abstract

A well-known formula of Whipple relates certain hypergeometric values $_7F_6(1)$ and $_4F_3(1)$. In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data $HD$, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data $HD$ are primitive and self-dual. If the data are also defined over $\mathbb Q$, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of $\ell$-adic representations of the absolute Galois group of $\mathbb Q$ attached to $HD$. For specialized choices of $HD$, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values $_7F_6(1)$ in Whipple's formula to the periods of these modular forms.