Isolated points on modular curves

Abstract: Faltings's theorem on rational points on subvarieties of abelian varieties can be used to show that all but finitely many algebraic points on a curve arise in families parametrized by $\mathbb{P}^1$ or positive rank abelian varieties; we call these finitely many exceptions isolated points. We study how isolated points behave under morphisms and then specialize to the case of modular curves. We show that isolated points on $X_1(n)$ push down to isolated points on a modular curve whose level is bounded by a constant that depends only on the j-invariant of the isolated point. This is joint work with A. Bourdon, O. Ejder, Y. Liu, and F. Odumodu.

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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