Isolated points on modular curves

TL;DR

This work generalizes the study of isolated points from the classical X1(X) setting to arbitrary modular curves X_H by introducing a robust single-source theorem: any isolated point with a given j-invariant on X_H arises from an isolated point on a single curve X_G determined by the elliptic curve’s adelic Galois image. It then develops a uniform, three-step method to locate all such isolated points across families of modular curves, combining finite Galois-image classifications with level-bounded reductions. The authors demonstrate the power of the approach through two applications: a complete classification of non-CM rational-j isolated points on all level-7 modular curves, and a conditional analysis for X_0(n) under a conjecture on Galois representations, including an explicit finite set of exceptional j-invariants. The results reveal a deep link between Mazur’s Program B and isolated points on modular curves, and provide practical tools (including a level-based isolation graph) for computing and bounding isolated points in new families. Overall, the paper offers a unifying, computationally tractable framework for identifying and characterizing rare isolated points across modular-curve families, with implications for Galois-representation images and uniformity questions in arithmetic geometry.

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-source 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 $\mathbb{Q}$.

Figures (3)

• Figure 1: The quotient of the isolation graph for modular curves of level 7 and closed points with $j$-invariant $\frac{3^{3} \cdot 5 \cdot 7^{5}}{2^{7}}$. Each vertex represents a set of closed points on such modular curves which have the same isolated-ness, while edges represent the relations of Theorems \ref{['thm:isolated_divisors:pullback_isolated_point']} and \ref{['thm:isolated_divisors:pushforward_isolated_point']}. The size of the nodes represents the degree of the closed points, while the color of the nodes indicates the genus of the underlying modular curve. The edges are colored according to the color of their source vertex. The graph is topologically sorted, so all edges go from top to bottom.
• Figure 2: The quotient of the isolation graph for modular curves of level 7 and closed points with $j$-invariant $\frac{3^{3} \cdot 5 \cdot 7^{5}}{2^{7}}$, as in Figure \ref{['fig:level_7:classes_isolation_graph']}. The subgraph induced by the possibly isolated vertices is highlighted in red, while all other vertices are not isolated by Theorem \ref{['thm:isolated_divisors:riemann_roch_criterion']}.
• Figure 3: The quotient of the isolation graph for modular curves of level 7 and closed points with $j$-invariant $\frac{3^{3} \cdot 5 \cdot 7^{5}}{2^{7}}$, as in Figure \ref{['fig:level_7:classes_isolation_graph']}. The subgraph induced by the closed points on geometrically connected modular curves is highlighted in red.

Theorems & Definitions (125)

• Theorem 1.1: Single-source theorem
• Example 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7: bourdon2019
• Theorem 1.8
• Remark 2.1
• Definition 2.2