An algebraic characterization of the Kronecker function

TL;DR

This paper provides an algebraic characterization of the Kronecker function by linking it to the fundamental solution of the Fay identity, and applies this to understand the generating series of extended period polynomials of Hecke eigenforms.

Contribution

It introduces a new algebraic characterization of the Kronecker function as the fundamental solution to the Fay identity, connecting it to period polynomials of Hecke eigenforms.

Findings

Characterization of the Kronecker function via the Fay identity

Description of generating series of extended period polynomials

Link between period relations and factorization properties

Abstract

We characterize the generating series of extended period polynomials of normalized Hecke eigenforms for $\operatorname{PSL}_2(\mathbb Z)$ studied by Zagier in terms of the period relations and existence of a suitable factorization. For this we prove a characterization of the Kronecker function as the `fundamental solution' to the Fay identity.