Families of Sporadic Points on Modular Curves

Abstract: A closed point $x$ on a curve $C$ is sporadic if there are only finitely many points of degree at most deg($x$). In the case where $C$ is the modular curve $X_1(N)$, a non-cuspidal sporadic point corresponds to an elliptic curve with a point of order $N$ defined over a number field of unusually low degree. In this talk, we will focus on sporadic points arising from $\mathbb{Q}$-curves, which are elliptic curves isogenous to their Galois conjugates. In particular, our investigations are inspired by the following question: Are there only finitely many non-CM $\mathbb{Q}$-curves which produce sporadic points on any modular curve of the form $X_1(N)$? I will show that an affirmative answer to this question would imply Serre's Uniformity Conjecture and discuss partial progress in the case of sporadic points of odd degree. This is joint work with Filip Najman.

number theory

Audience: researchers in the topic

( slides | video )

Rational Points and Galois Representations

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