Hurwitz class numbers with level and modular correspondences

TL;DR

This paper derives formulas for Hurwitz class numbers at various levels by analyzing intersection numbers of modular correspondences, extending Atkin-Lehner involutions to compute these intersections at cusps.

Contribution

It introduces a method to compute Hurwitz class numbers for levels with genus zero using intersection theory and generalized Atkin-Lehner involutions.

Findings

Formulas for Hurwitz class numbers at specific levels

A new approach to intersection multiplicities at cusps

Extension of Atkin-Lehner involutions to new subgroup settings

Abstract

In this paper, we prove Hurwitz-Eichler type formulas for Hurwitz class numbers with each level $ M $ when the modular curve $ X_0(M) $ has genus zero. A key idea is to calculate intersection numbers of modular correspondences with the level in two different ways. A generalization of Atkin-Lehner involutions for $ \Gamma_0(M) $ and its subgroup $ \Gamma_0^{(M')}(M) $ is introduced to calculate intersection multiplicities of modular correspondences at cusps.