Ideal class groups of number fields associated to modular Galois representations

2022/05/11 by Naoto Dainobu, Dainobu, Naoto
Mathematics · #11G05 (Primary) #11R29 #11R34 (Secondary) #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2205.05238

openalex publication_date 2022/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime number and f a modular form. We consider the \mathbbF_p-valued Galois representation ρ_f attached to f and its twist ρ_{f, D} by the quadratic character χ_D corresponding to a quadratic discriminant D. We define K_{f, D} to be the field corresponding to the kernel of ρ_{f, D}. In this article, we investigate the ideal class group Cl(K_{f, D}) of the number field K_{f, D} as a Gal(K_{f, D}/ℚ)-module. We give a condition which implies the existence of a Gal(K_{f, D}/ℚ)-equivariant surjective homomorphism from Cl(K_{f, D})⊗ \mathbbF_p to the representation space M_{f, D} of ρ_{f, D}, using Bloch and Kato's Selmer group of ρ_{f, D}. We also give some numerical examples where we have such surjections by calculating the central value of the L-function of f twisted by χ_D under Bloch and Kato's conjecture. Our main result in this paper is a partial generalization of the previous result of Prasad and Shekhar on elliptic curves to higher weight modular forms.

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