Ideal class groups of number fields associated to modular Galois representations

TL;DR

This paper explores the structure of ideal class groups of certain number fields linked to modular Galois representations, providing conditions for surjections onto representation spaces and extending previous elliptic curve results to higher weight forms.

Contribution

It generalizes prior work on elliptic curves to higher weight modular forms, establishing new conditions for class group surjections related to modular Galois representations.

Findings

Identifies conditions for Galois-equivariant surjections from class groups to representation spaces.

Provides numerical examples supporting the theoretical results.

Extends results from elliptic curves to modular forms of higher weight.

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.