The $2$-Selmer group of a number field and heuristics for narrow class groups and signature ranks of units

TL;DR

This paper studies the 2-Selmer group of a number field through a homomorphism called the 2-Selmer signature map, providing theoretical predictions and computational evidence for the distribution of narrow class groups and unit signature ranks.

Contribution

It introduces the 2-Selmer signature map, proves its image is a maximal totally isotropic subspace, and applies this to predict distributions of class groups and units, supported by extensive computations.

Findings

The 2-Selmer signature map's image is a maximal totally isotropic subspace.

Predictions on the density of fields with specific class group 2-rank.

Computational results match theoretical predictions very well.

Abstract

We investigate in detail a homomorphism which we call the 2-Selmer signature map from the $2$-Selmer group of a number field $K$ to a nondegenerate symmetric space, in particular proving the image is a maximal totally isotropic subspace. Applications include precise predictions on the density of fields $K$ with given narrow class group 2-rank and with given unit group signature rank. In addition to theoretical evidence, extensive computations for totally real cubic and quintic fields are presented that match the predictions extremely well. In an appendix with Richard Foote, we classify the maximal totally isotropic subspaces of orthogonal direct sums of two nondegenerate symmetric spaces over perfect fields of characteristic 2 and derive some consequences, including a mass formula for such subspaces.