The Elliptic Hypergeometric Functions Associated to the Configuration Space of Points on an Elliptic Curve I : Twisted Cycles

TL;DR

This paper studies elliptic hypergeometric functions linked to point configurations on elliptic curves, calculating twisted homology groups and connection matrices to understand their analytic continuation properties.

Contribution

It introduces new methods for computing twisted homology and intersection forms for elliptic hypergeometric functions, extending classical hypergeometric theory to elliptic curves.

Findings

Computed twisted homology groups for elliptic hypergeometric functions

Derived intersection forms and connection matrices

Enhanced understanding of analytic continuation of elliptic hypergeometric functions

Abstract

We consider the Euler type integral associated to the configuration space of points on an elliptic curve, which is an analogue of the hypergeometric function associated to the configuration space of points on a projective line. We calculate the {\it twisted homology group}, with coefficients in the local system associated to a power function $g^{\alpha}$ of an elliptic function $g$, and the intersection form. Applying these calculations, we describe the {\it connection matrices} representing the linear isomorphisms induced from analytic continuations of the functions defined by the integrations of $g^{\alpha}$ over twisted cycles.