Sporadic points of odd degree on $X_1(N)$ coming from $\mathbb{Q}$-curves

TL;DR

This paper investigates sporadic points of odd degree on modular curves $X_1(N)$ arising from $Q$-curves, establishing finiteness results and connections to Serre's Uniformity Conjecture.

Contribution

It proves all non-CM $Q$-curves producing sporadic points of odd degree are in the isogeny class of a specific elliptic curve with $j$-invariant $-140625/8$, and links a stronger finiteness to Serre's conjecture.

Findings

All non-CM $Q$-curves with sporadic points of odd degree are in the isogeny class of a specific elliptic curve.

A stronger finiteness statement would imply Serre's Uniformity Conjecture.

Most known sporadic points come from CM elliptic curves.

Abstract

We say a closed point $x$ on a curve $C$ is sporadic if there are only finitely many points on $C$ of degree at most deg$(x)$. In the case where $C$ is the modular curve $X_1(N)$, most known examples of sporadic points come from elliptic curves with complex multiplication (CM). We seek to understand all sporadic points on $X_1(N)$ corresponding to $\mathbb{Q}$-curves, which are elliptic curves isogenous to their Galois conjugates. This class contains not only all CM elliptic curves, but also any elliptic curve $\overline{\mathbb{Q}}$-isogenous to one with a rational $j$-invariant, among others. In this paper, we show that all non-CM $\mathbb{Q}$-curves giving rise to a sporadic point of odd degree lie in the $\overline{\mathbb{Q}}$-isogeny class of the elliptic curve with $j$-invariant $-140625/8$. In addition, we show that a stronger version of this finiteness result would imply Serre's Uniformity Conjecture.