Weierstrass points on certain modular groups (Q897566)

Let \(N\) be a natural number, \(\Delta\) a subgroup of \((\mathbb Z/N\mathbb Z)^{\ast}\), \(\Gamma_{\Delta}(N)\) the subgroup \[ \Big\{ {a\, b\choose c\,d} | c \equiv 0\,(\bmod N),(a \bmod N) \in \Delta \Big\} \text{ of } \text{SL}_2(\mathbb Z), \] and \(X_{\Delta}(N)\) the corresponding modular curve. The authors assume that \(N\) is divisible by the square of a prime number \(p\) and investigate Weierstrass points on \(X_{\Delta}(N)\). The main results are Theorems 2.8, 2.9 and 2.11, which present sufficient conditions (on \(\Delta\) and the nature of \(M:= N/p^2\)) for the cusp \(0\) of \(X_{\Delta}(N)\) to be a Weierstrass point.