On Siegel's modular curve of level 5 and the class number one problem
Given a prime \(p\), let \(z\) be an element of \(\mathbb P (\mathbb F_{p^2})\) with \(z \notin \mathbb P (\mathbb F_p)\), and let \(N_z\) be the stabilizer of \(\{ z. z^p \}\) in \(GL_2 (\mathbb F_p)\). If \(X(p)\) denotes the modular curve which classifies isomorphism classes of elliptic curves with level \(p\) structure, then \(X^+_{\text{ns}} (p)\) is the quotient of \(X(p)\) by \(N_z \subset GL_2 (\mathbb F_p)\). An explicit parametrization of \(X^+_{\text{ns}} (5)\) was obtained by \textit{C. L. Siegel} [Invent. Math. 5, 180-191 (1968; Zbl 0175.33602)] and was used to solve the class number one problem. In this paper the author derives another explicit parametrization of the modular curve \(X^+_{\text{ns}} (5)\) over \(\mathbb Q\).
scientific article; zbMATH DE number 1271092
scientific article