Rational points on modular curves: parameterization and geometric explanations

TL;DR

This paper, assuming a conjecture, provides a new parameterization of rational points on modular curves, linking them to finitely many other curves and confirming a geometric philosophy about their origins.

Contribution

It refines Zywina's work to give a conditional parameterization of Galois representations and rational points on modular curves using finitely many curves.

Findings

Identifies 41 special elliptic curves with fixed Galois images.

Provides an explicit parameterization of rational points on all modular curves.

Confirms that all rational points arise from the geometry of modular curves.

Abstract

We show that, conditional on Zywina's effective version of the Serre uniformity conjecture, there is a natural way to parameterize non-CM $\mathbb{Q}$-rational points on all modular curves in terms of the rational points on finitely many modular curves. Our proof refines Zywina's work to give a (conditional) parameterization of the images of adelic Galois representations of elliptic curves. In particular, we show that there are 41 $j$-invariants of elliptic curves whose associated Galois image does not vary in an infinite family. Using our explicit parameterization, we show that all rational points on all modular curves arise from the geometry of modular curves in a formal sense, confirming a philosophy of Mazur and Ogg.