Applications of Kronecker's limit formula for elliptic Eisenstein series

TL;DR

This paper explores the use of Kronecker's limit formula in elliptic Eisenstein series to derive modular form factorizations and prove Weil's reciprocity law, with explicit examples for moonshine and congruence groups.

Contribution

It introduces new applications of Kronecker's limit formula to factorize modular forms and establish Weil's reciprocity, including explicit computations for specific groups.

Findings

Factorization of holomorphic modular forms achieved

Proof of Weil's reciprocity law provided

Explicit Kronecker limit functions computed for certain groups

Abstract

We develop two applications of the Kronecker's limit formula associated to elliptic Eisenstein series: A factorization theorem for holomorphic modular forms, and a proof of Weil's reciprocity law. Several examples of the general factorization results are computed, specifically for certain moonshine groups, congruence subgroups, and, more generally, non-compact subgroups with one cusp. In particular, we explicitly compute the Kronecker limit function associated to certain elliptic points for a few small level moonshine groups.