Kronecker limit formulas and scattering constants for Fermat curves

TL;DR

This paper derives Kronecker limit formulas for non-congruence subgroups related to Fermat curves and uses these to compute their scattering constants, advancing spectral theory in hyperbolic geometry.

Contribution

It provides the first Kronecker limit formulas for Fermat curve-associated non-congruence subgroups and calculates their scattering constants.

Findings

Kronecker limit formula for Fermat curve subgroups established

Scattering constants for Fermat curves explicitly computed

Enhances understanding of spectral properties of non-congruence subgroups

Abstract

Eisenstein series are real analytic functions which play a central role in spectral theory of the hyperbolic Laplacian. Kronecker limit formulas determine their connection to modular forms. The main result of this work is Theorem 7.2 in which a Kronecker limit formula for a family of non-congruence subgroups associated with the Fermat curves is presented. As an application we can determine the scattering constants for the Fermat curves in Theorem 8.1.