Isolated points on modular curves

Abstract

We study isolated points on the modular curves $X_{H}$, for $H$ a subgroup of $\operatorname{GL}_{2}(\mathbb{Z}/n \mathbb{Z})$ for some $n \geq 1$. In particular, we prove a single-sink theorem for such isolated points, which traces the existence of all such isolated points with the same $j$-invariant back to an isolated point on a single curve. Building on this result, we also present a uniform strategy for determining the isolated points on any family of modular curves. As an example, we use this strategy to classify the isolated points with rational $j$-invariant on all modular curves of level 7, as well as the modular curves $X_{0}(n)$, the latter assuming a conjecture on images of Galois representations of elliptic curves over $\mathbb{Q}$. Underpinning all of this, we develop a theory of isolated divisors on geometrically disconnected varieties, which may be of independent interest.