The rational cuspidal divisor class group of X₀(N)

2019/08/18 by Hwajong Yoo, Yoo, Hwajong · 2 citations
Mathematics · #11G16 #11G18 #14G05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1908.06411

openalex publication_date 2019/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any positive integer N, we completely determine the structure of the rational cuspidal divisor class group of X₀(N), which is conjecturally equal to the rational torsion subgroup of J₀(N). More specifically, for a given prime ℓ, we construct a rational cuspidal divisor Z_ℓ(d) for any non-trivial divisor d of N. Also, we compute the order of the linear equivalence class of the divisor Z_ℓ(d) and show that the ℓ-primary subgroup of the rational cuspidal divisor class group of X₀(N) is isomorphic to the direct sum of the cyclic subgroups generated by the linear equivalence classes of the divisors Z_ℓ(d).

Cited by

• Reducibility and rational torsion in modular abelian varieties
• On Some Open Cases of a Conjecture of Conrad, Edixhoven and Stein

Related

• The rational cuspidal subgroup of J₀(N)
• The rational torsion subgroup of J₀(N)
• Modular unit and cuspidal divisor class groups of X₁(N)
• The rational cuspidal subgroup of J₀(p²M) with M squarefree
• The rational torsion subgroups of Drinfeld modular Jacobians and Eisenstein pseudo-harmonic cochains