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

TL;DR

This paper classifies points on certain hyperelliptic modular curves over quadratic fields, revealing that most elliptic curves with specific isogenies are Q-curves, and describes their relation to Galois conjugates and twists.

Contribution

It determines all quadratic points on hyperelliptic $X_0(n)$ curves with rank zero Jacobians and provides a detailed classification of elliptic curves with $n$-isogenies over quadratic fields.

Findings

Most elliptic curves with $n$-isogenies over quadratic fields are Q-curves.

Finitely many exceptions are explicitly listed.

Every such Q-curve is related to its Galois conjugate via quadratic twists.

Abstract

Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ 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.