Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series

TL;DR

This paper generalizes Dedekind symbols and Eisenstein series to Fuchsian groups, establishing their properties and showing higher-order Dedekind symbols for certain groups are rational, extending classical modular form results.

Contribution

It introduces modular Dedekind symbols for Fuchsian groups and higher-order Eisenstein series, expanding the classical theory beyond the modular group and analyzing their properties.

Findings

Higher-order Dedekind symbols are rational for genus one congruence groups.

The paper extends Kronecker limit formulas to more general Fuchsian groups.

Properties of these symbols are detailed through limit formulas.

Abstract

Let $E(z,s)$ be the non-holomorphic Eisenstein series for the modular group $SL(2,{\mathbb Z})$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $s=1$ of $E(z,s)$ is essentially the logarithm of the Dedekind eta function. This eta function is a weight $1/2$ modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups $\Gamma$. The analogue of the eta function has a multiplier system that may be expressed in terms of a map $S:\Gamma \to {\mathbb R}$ which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from $\Gamma$ to ${\mathbb C}$ produces higher-order Eisenstein series. These series share many of the properties of $E(z,s)$ though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher-order Dedekind symbols $S^*:\Gamma \to {\mathbb R}$. As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups $\Gamma_0(N)$ are rational.