Degrees of points with rational $j$-invariant on $X_{0}(n)$ and $X_{1}(n)$

TL;DR

This paper classifies the degrees of rational $j$-invariant points on modular curves $X_0(n)$ and $X_1(n)$, determining which occur infinitely often unconditionally and which finitely often under a conjecture, with applications to isolated $j$-invariants.

Contribution

It introduces the concept of $ ext{H}$-closures of subgroups of $ ext{GL}_2( ext{Zhat})$ and computes these closures for Galois representations, advancing understanding of rational points on modular curves.

Findings

Classified degrees of rational $j$-invariant points on $X_0(n)$ and $X_1(n)$.

Computed $ ext{B}_0(n)$- and $ ext{B}_1(n)$-closures of Galois representations.

Identified infinitely and finitely occurring degrees, with an application to isolated $j$-invariants.

Abstract

We give a classification of the degrees of the points with rational $j$-invariant on the modular curves $X_{0}(n)$ and $X_{1}(n)$. The degrees which occur infinitely often are computed unconditionally, while those which occur finitely often are determined assuming a conjecture of Zywina. To achieve this, we define the notion of $\mathcal{H}$-closures of subgroups of $\operatorname{GL}_{2}(\widehat{\mathbb{Z}})$, and compute the $\mathcal{B}_{0}(n)$- and $\mathcal{B}_{1}(n)$-closures of images of Galois representations of elliptic curves defined over $\mathbb{Q}$. An application to computing the set of isolated $j$-invariants in $\mathbb{Q}$ is also given.