Bielliptic intermediate modular curves (Q2318388)

Let \(N\) be a positive integer and \(\Delta\) be a subgroup of \((\mathbb Z/N\mathbb Z)^*\) containing \(\{\pm 1\}\). Let \(\Gamma_\Delta(N)\) be the subgroup of \(\Gamma_0(N)\) consisting of matrices \(\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right), a\mod N\in \Delta\) and \(X_\Delta\) be the modular group associated with \(\Gamma_\Delta\). Then \(X_\Delta\) is an intermediate curve between the modular curves \(X_1(N)\) and \(X_0(N)\). The problems considered in this article are concerned with determining which of \(X_\Delta\) are bielliptic, and with finding all \(X_\Delta\) having infinitely many quadratic points over \(\mathbb Q\) where \(\Delta\ne\{\pm 1\}, (\mathbb Z/N\mathbb Z)^*\), because these problems have been solved for the curves \(X_1(N) (\Delta=\{\pm 1\})\) and \(X_0(N) (\Delta=(\mathbb Z/N\mathbb Z)^*)\). Here, bielliptic curve is a curve \(X\) with a bielliptic involution \(\sigma\) such that the quotient \(X/\langle\sigma\rangle\) is elliptic. All level \(N\) such that \(X_0(N)\) is rational,elliptic,hyperelliptic or bielliptic have been determined. Therefore, using the fact that \(X_0(N)\) is rational,elliptic, hyperelliptic or bielliptic when \(X_\Delta\) is bielliptic, the authors list all possible \((N,\Delta)\) such that \(X_\Delta\) could be bielliptic. By determining the fixed points of an involution \(W\) of \(X_\Delta\) constructed from elements of \(\Gamma_0(N)\) and Atkin-Lehner involutions of \(X_0(N)\) and further applying Hurwitz formula, they determine whether \(W\) is bielliptic or not. As an effective tool to exclude non-bielliptic curves from candidates, they use Castelnuvo's inequality which gives an inequality among the genus of a function field and those of its subfields. The authors determine all \(25\) bielliptic curves \(X_\Delta\) and their bielliptic involutions. For quadratic points, they show that the only modular curve \(X_\Delta\) of genus \(>1\) which has infinitely many quadratic points over \(\mathbb Q\) is the unique hyperelliptic curve \(X_{\Delta_1}(21)\) where \(\Delta_1=\{\pm 1,\pm8\}\). Many suggestive examples are included.