$\ell$-Torsion in Class Groups via Dirichlet $L$-functions

TL;DR

This work develops an approach to bound the $\ell$-torsion $h_{\ell}(K)$ of class groups for degree 2 and 3 fields without relying on GRH. It leverages Dirichlet $L$-functions and zero-free regions, using Gaussian weightings and contour integrals to connect $h_{\ell}(K)$ to small-norm prime ideals and $L$-function data, thereby overcoming obstacles from exceptional zeros. The main results show unconditional bounds $h_{\ell}(K) \ll |\Delta_K|^{1/2-1/(2\ell)+\varepsilon}$ for quadratic fields and $h_{\ell}(K) \ll |\Delta_K|^{1/2-1/(4\ell)+\varepsilon}$ for cubic fields under zero-free assumptions, with effective corollaries for smooth discriminants and q-analogs; a complementary zero-free-region theorem (Theorem ['zfr']) clarifies when such bounds are effective. The results contribute to the broader effort to beat the trivial Landau bound in low-degree families and illuminate the role of $L$-function analytic properties in arithmetic statistics of class groups.

Abstract

For a prime $\ell$, let $h_\ell(K)$ denote the $\ell$-part of the class number of the number field $K$. We investigate upper bounds for $h_\ell(K)$ when $K$ is quadratic or cubic, particularly in the case in which the discriminant of $K$ is smooth. This is achieved using properties of Dirichlet $L$-functions.

Theorems & Definitions (12)

• Theorem 1
• Lemma 2
• Lemma 3
• Theorem 4
• Theorem 5
• Theorem 6
• Corollary 7
• Theorem 8
• Corollary 9
• Theorem 10