Weierstrass points at cusps on special modular curves
Let \(X_0(N)\) be the complete modular curve of level \(N\), let \(g(N)\) be its genus and let \(S_2(N)\) be the space of cusp forms of weight 2 and level \(N\) (which may be naturally identified with the space of holomorphic differential forms on \(X_0(N)\)). Recall that the cusp \(\infty\) of \(X_0(N)\) is a Weierstrass point if and only if there exists \(f \in S_2(N)\) such that \(\text{ord}_\infty (f) > g(N)\), and let \({\mathcal W} S_2(N) = \{ f \in S_2(N)\mid \text{ord}_\infty f > g(N) \}\). Let \(\ell\) be a prime and \(\mathbb{Z}_{(\ell)}\) be the localization of \(\mathbb{Z}\) at \(\ell\); for a power series \(f = \sum_{n \geq 0} a_f(n) q^n \in \mathbb{Z}_{(\ell)}q\) with \(\ell\)-integral coefficients, denote by \(\overline{f} = \sum_{n \geq 0} \overline{a_f(n)} q^n \in \mathbb{F}_\ell q\) the reduction of \(f\) modulo \(\ell\), and finally define \({\mathcal W}\overline{{\mathcal L}_2(N)} = \{ \overline{f} \mid f \in {\mathcal L}_2(N), \text{ord}_\infty \bar{f} > g(N)\}\), where \({\mathcal L}_2(N)\) is the \(\mathbb{Z}\)-module of lattice forms in \(S_2(N)\) with integral Fourier coefficients. In the present work, the author studies the space \({\mathcal W}\overline{{\mathcal L}_2(N)}\) in the case where \(N = 11 p \), where \(p > 11\) is a prime satisfying \(p \equiv 1\pmod 5\). Among other results it is proved that \(\dim_{\mathbb{F}_p} {\mathcal W}\overline{{\mathcal L}_2(11p)} = 1\); moreover, if there exists a non-zero form \(f \in {\mathcal W}S_2(11 p)\) (which we may suppose that has integral Fourier coefficients and is non-zero modulo \(p\)) then \(f\) is proportional to a meromorphic modular function of weight 2 and level \(11p\) specified in the text.
scientific article; zbMATH DE number 2063947
scientific article