Hurwitz class numbers with level and modular correspondences

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 ${\Gamma }_0^{(M^{\prime })}(M)$ is introduced to calculate intersection multiplicities of modular correspondences at cusps.

29 pages, 7 tables