Isolated points on modular curves of prime-power level

2025/12/12 by Chris Calger, Calger, Chris
Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2512.11996

openalex publication_date 2025/12/12 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28

Abstract

An isolated point on an algebraic curve is a closed point not belonging to a collection of points of the same degree parametrized by P¹ or a positive rank abelian subvariety of the curve's Jacobian. We study the sets of j-invariants, in extensions of bounded degree, that arise as the j-invariant of an isolated point on a modular curve. We obtain finiteness results on these sets for families of modular curves with prime-power level. This is related to recent work of Bourdon and Ejder, who classified rational j-invariants of isolated points on the families X₁(n) and X₀(n), for n a prime power.

Citations

• Isolated points on modular curves
• Effective bounds for adelic Galois representations attached to elliptic curves over the rationals
• Propri�t�s galoisiennes des points d'ordre fini des courbes elliptiques

Related

• Isolated j-invariants arising from the modular curve X₀(n)
• Rational isolated j-invariants from X₁(ℓⁿ) and X₀(ℓⁿ)
• Isolated Points on X₁(ℓⁿ) with rational j-invariant
• Isolated points on modular curves
• On the level of modular curves that give rise to isolated j-invariants