On rational Eisenstein primes and the rational cuspidal groups of\n modular Jacobian varieties

2015/10/11 by Hwajong Yoo, Yoo, Hwajong
Mathematics · #11F33 #11F80 #11G18 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1510.03016

openalex publication_date 2015/10/11 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

Let N be a non-squarefree positive integer and let \ℓ be an odd prime\nsuch that \ℓ² does not divide N. Consider the Hecke ring mathbbT(N)\nof weight 2 for \Γ₀(N), and its rational Eisenstein primes of\n mathbbT(N) containing \ℓ, defined in Section 3. If mathfrakm is\nsuch a rational Eisenstein prime, then we prove that mathfrakm is of the\nform (\ℓ, ~\I^D_{M, N}), where the ideal \I^D_{M, N} of\n mathbbT(N) is also defined in Section 3. Furthermore, we prove that\n\C(N)[ mathfrakm] \≠ 0, where \C(N) is the rational\ncuspidal group of J₀(N). To do this, we compute the precise order of the\ncuspidal divisor \C^D_{M, N}, defined in Section 4, and the index of\n\I^D_{M, N} in mathbbT(N)\⊗ \ℤ_\ℓ.\n

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