Differential and Functional Identities for the Elliptic Trilogarithm

TL;DR

This paper generalizes classical elliptic function identities to derivatives of the elliptic trilogarithm, deriving new functional and differential identities that facilitate elliptic solutions to associativity equations.

Contribution

It introduces generalized Frobenius-Stickelberger identities involving derivatives of the elliptic trilogarithm, expanding the theoretical framework of elliptic functions.

Findings

Derived a differential identity for the elliptic trilogarithm.

Established generalized Frobenius-Stickelberger identities.

Connected identities to elliptic solutions of associativity equations.

Abstract

When written in terms of $\vartheta$-functions, the classical Frobenius-Stickelberger pseudo-addition formula takes a very simple form. Generalizations of this functional identity are studied, where the functions involved are derivatives (including derivatives with respect to the modular parameter) of the elliptic trilogarithm function introduced by Beilinson and Levin. A differential identity satisfied by this function is also derived. These generalized Frobenius-Stickelberger identities play a fundamental role in the development of elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde equations of associativity, with the simplest case reducing to the above mentioned differential identity.