Ideal class groups of number fields associated to modular Galois representations

Abstract

Let $p$ be an odd prime number and $f$ a modular form. We consider the $\mathbb{F}_p$-valued Galois representation $\bar{\rho}_f$ attached to $f$ and its twist $\bar{\rho}_{f, D}$ by the quadratic character $\chi_D$ corresponding to a quadratic discriminant $D$. We define $K_{f, D}$ to be the field corresponding to the kernel of $\bar{\rho}_{f, D}$. In this article, we investigate the ideal class group $\mathrm{Cl}(K_{f, D})$ of the number field $K_{f, D}$ as a $\mathrm{Gal}(K_{f, D}/\mathbb{Q})$-module. We give a condition which implies the existence of a $\mathrm{Gal}(K_{f, D}/\mathbb{Q})$-equivariant surjective homomorphism from $\mathrm{Cl}(K_{f, D})\otimes \mathbb{F}_p$ to the representation space $M_{f, D}$ of $\bar{\rho}_{f, D}$, using Bloch and Kato's Selmer group of $\bar{\rho}_{f, D}$. We also give some numerical examples where we have such surjections by calculating the central value of the $L$-function of $f$ twisted by $\chi_D$ 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.