Hyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fields

Abstract

Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge 2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb{Q}$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to $\overline{\mathbb{Q}}$-isomorphism, all but finitely many elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb{Q}$-curves, and we list all exceptions. We also show that, again with finitely many exceptions up to $\overline{\mathbb{Q}}$-isomorphism, every $\mathbb{Q}$-curve $E$ over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by some quadratic extension $L$ of $K$; we determine $d$ and $L$ explicitly.

30 pages, published version displays a wrong equation of X_0(26), has a few missing exceptional points on X_0(n) for n=22,26 and 46, three points too many on X_0(23), and a typo in a point for X_0(29). Magma code is included

Cited by in corpus (13)

• Computing images of Galois representations attached to elliptic curves
• Fermat's Last Theorem and modular curves over real quadratic fields
• Explicit isogenies of prime degree over quadratic fields
• Computing quadratic points on modular curves $X_0(N)$
• Computing representation matrices for the action of Frobenius to cohomology groups
• Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields
• Quadratic Points on Non-Split Cartan Modular Curves
• On elliptic curves with $p$-isogenies over quadratic fields
• Computing points on bielliptic modular curves over fixed quadratic fields
• Classification of torsion of elliptic curves over quartic fields
• Fields generated by points on superelliptic curves
• Explicit isogenies of prime degree over number fields
• Fermat's Last Theorem over $\mathbb{Q}(\sqrt{2},\sqrt{3})$