On the level of modular curves that give rise to isolated $j$-invariants

TL;DR

This paper investigates sporadic and isolated points on modular curves $X_1(N)$, establishing bounds on their associated degrees and demonstrating finiteness of such points with bounded $j$-invariants.

Contribution

It proves that sporadic and isolated points on $X_1(N)$ project to points on curves with degrees bounded by constants depending on $j$-invariants, with conditional finiteness results.

Findings

Non-CM sporadic and isolated points map to points on curves with bounded degree.

Conditional bounds depend only on the degree of the $j$-invariant.

Finiteness of $j$-invariants for bounded degrees is established under certain conditions.

Abstract

We say a closed point $x$ on a curve $C$ is sporadic if $C$ has only finitely many closed points of degree at most $\operatorname{deg}(x)$ and that $x$ is isolated if it is not in a family of effective degree $d$ divisors parametrized by $\mathbb{P}^1$ or a positive rank abelian variety (see Section 4 for more precise definitions and a proof that sporadic points are isolated). Motivated by well-known classification problems concerning rational torsion of elliptic curves, we study sporadic and isolated points on the modular curves $X_1(N)$. In particular, we show that any non-cuspidal non-CM sporadic, respectively isolated, point $x \in X_1(N)$ maps down to a sporadic, respectively isolated, point on a modular curve $X_1(d)$, where $d$ is bounded by a constant depending only on $j(x)$. Conditionally, we show that $d$ is bounded by a constant depending only on the degree of $\mathbb{Q}(j(x))$, so in particular there are only finitely many $j$-invariants of bounded degree that give rise to sporadic or isolated points.