Kronecker limit formulas and scattering constants for Fermat curves

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.