Ideal class groups and torsion in Picard groups of varieties

TL;DR

This paper introduces a new technique linking the Picard groups of varieties over  to the construction of number fields with large ideal class groups, providing a unified framework and improved quantitative results.

Contribution

It develops a general method to construct and count number fields with nontrivial m-rank class groups via specialization of Picard groups of varieties.

Findings

Reduces the problem of constructing large-rank class groups to finding hyperelliptic curves with specific properties.

Reinterprets previous results within the new framework.

Provides the best known quantitative bounds for large-rank class groups in number fields.

Abstract

We give a new general technique for constructing and counting number fields with an ideal class group of nontrivial m-rank. Our results can be viewed as providing a way of specializing the Picard group of a variety V over $\mathbb{Q}$ to obtain class groups for number fields $\mathbb{Q}(P)$, $P\in V(\Qbar)$, for certain families of points P. In particular, we show how the problem of constructing quadratic number fields with a large-rank ideal class group can be reduced to the problem of finding a hyperelliptic curve with a rational Weierstrass point and a large rational torsion subgroup in its Jacobian. Furthermore, we show how many previous results on constructing large-rank ideal class groups can be fit into our framework and rederived. As an application of our technique, we derive a quantitative version of a theorem of Nakano. This gives the best known general quantitative result on number fields with a large-rank ideal class group.