Generalized Verlinde formulae for some Riemann surface automorphisms

TL;DR

This paper derives generalized Verlinde formulae for automorphisms of specific Riemann surfaces, linking modular representations to automorphic properties and providing congruence relations for these representations.

Contribution

It introduces new generalized Verlinde formulae for certain algebraic curves and explores their implications for modular representations and automorphic properties.

Findings

Derived generalized Verlinde formulae for specific Riemann surfaces.

Established congruence relations on modular representations.

Illustrated the connection between arithmetic properties and automorphic features.

Abstract

The generalized Verlinde formulae expressing traces of mapping classes corresponding to automorphisms of certain Riemann surfaces, and the congruence relations on allowed modular representations following from them are presented. The surfaces considered are families of algebraic curves given by suitably chosen equations, the modular curve $\mathcal{X}(11)$, and a factor curve of $\mathcal{X}(8)$. The examples of modular curves illustrate how the study of arithmetic properties of suitable modular representations can be used to gain information on automorphic properties of Riemann surfaces.