Special function identities from superelliptic Kummer varieties

TL;DR

This paper reveals a geometric origin for special function identities by constructing a generalized Kummer variety from superelliptic curves, leading to a multivariate generalization of Clausen's Formula.

Contribution

It introduces a geometric construction linking hypergeometric identities to superelliptic Kummer varieties, bridging algebraic transformations and nodal surface theory.

Findings

Factorization of Appell's hypergeometric series has a geometric origin.

Constructed a generalized Kummer variety from superelliptic curves.

Derived a multivariate generalization of Clausen's Formula.

Abstract

We prove that the factorization of Appell's generalized hypergeometric series satisfying the so-called quadric property into a product of two Gauss' hypergeometric functions has a geometric origin: we first construct a generalized Kummer variety as minimal nonsingular model for a product-quotient surface with only rational double points from a pair of superelliptic curves of genus $2r-1$ with $r \in \mathbb{N}$. We then show that this generalized Kummer variety is equipped with two fibrations with fibers of genus $2r-1$. When periods of a holomorphic two-form over carefully crafted transcendental two-cycles on the generalized Kummer variety are evaluated using either of the two fibrations, the answer must be independent of the fibration and the aforementioned family of special function identities is obtained. This family of identities can be seen as a multivariate generalization of Clausen's Formula. Interestingly, this paper's finding bridges Ernst Kummer's two independent lines of research, algebraic transformations for the Gauss' hypergeometric function and nodal surfaces of degree four in $\mathbb{P}^3$.