Estimation of genus of arithmetic curves and applications

Abstract

For any element γ∈Γ0(N) and a positive integer N, we find the genus of arithmetic curve 〈Γ1(N),γΦ〉\ℋ*, where \(\Phi=\bigl(\begin{scriptsize}\begin{array}{c@{\ }c}0&-1\\N&0\end{array}\end{scriptsize}\bigr)\) is the Fricke involution. We obtain that the genus of 〈Γ1(N),γΦ〉\ℋ* is zero if and only if 1≤N≤12 or N=14, 15. As its applications, since the genus formula is independent of γ, we determine the Hauptmoduln for the groups 〈Γ1(N),Φ〉 of genus zero which will be used to generate appropriate ray class fields over imaginary quadratic fields, and show that the fixed point of γΦ in ℋ is a Weierstrass point of Γ1(N) for all but finitely many N, which is a direct generalization of Lehner-Newman’s use of Schoeneberg’s Theorem.