Bielliptic intermediate modular curves

Abstract

We determine which of the modular curves $X_{\Delta }(N)$, that is, curves lying between $X_0(N)$ and $X_1(N)$, are bielliptic. Somewhat surprisingly, we find that one of these curves has exceptional automorphisms. Finally we find all $X_{\Delta }(N)$ that have infinitely many quadratic points over $\mathbb{Q}$.

35 pages, published version, improved exposition, added many details

References in corpus (1)

• Normalizers of Congruence Groups in $S{L}_2(\mathbb{R})$ and Automorphisms of Lattices

Cited by in corpus (5)

• Bielliptic quotient modular curves of $X_0(N)$
• Odd degree isolated points on $X_1(N)$ with rational $j$-invariant
• Intermediate modular curves with infinitely many cubic points over $\mathbb{Q}$
• Bielliptic modular curves $X_0^{\ast }(N)$ with square-free levels
• Hyperelliptic and trigonal modular curves in characteristic $p$