On scattering constants for a non-congruence subgroup

TL;DR

This paper derives explicit formulas for scattering constants linked to a non-congruence subgroup, using Belyi maps of elliptic curves, advancing understanding of special values of associated Dirichlet series.

Contribution

It provides the first closed-form expressions for scattering constants of a non-congruence subgroup derived from elliptic curves via Belyi maps.

Findings

Closed formulas for scattering constants are obtained.

The formulas relate scattering constants to elliptic curve invariants.

This work enhances the understanding of non-congruence subgroups and their associated Dirichlet series.

Abstract

Scattering constants are special values of Dirichlet series associated to non-holomorphic Eisenstein series. In this paper we give closed formulas for the scattering constants related to a non-congruence subgroup obtained via a Belyi map of an elliptic curve.