Minimal resolution of Atkin–Lehner quotients of X0(N)

Let X0(N) be the classic modular curve of level N over Z. Let WM be the Atkin–Lehner involution of X0(N) associated to a divisor M with (M,N/M)=1. In this paper an explicit description is given for the minimal resolution over Z[1/6] of the Atkin–Lehner quotient X0(N)/WM. As an application a new proof of Deuring's formula on the number of supersingular j-invariants in Fp is given. In certain cases it is also shown that the action of Hecke operators on the component group of the Jacobian of the Atkin–Lehner quotient is Eisenstein.