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')}(M)} $ is introduced to calculate intersection multiplicities of modular correspondences at cusps.