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.

24pages