Archimedean aspects of the Cohen-Lenstra heuristics

Abstract: Like rational points on elliptic curves, units in number rings are gems of arithmetic. Refined questions about units remain difficult to answer, often embedded within difficult questions about class groups. For example: how often in a number ring is it that all totally positive units are squares?

Absent theorems, we may still try to predict the answer to these questions. In this talk, we present heuristics (and some theorems!) for signatures of unit groups, inspired by the Cohen-Lenstra heuristics, formulating precise conjectures and providing evidence for them. A key role is played by a lustrous structure of number rings we call the 2-Selmer signature map. This structure clarifies the provenance of reflection theorems, like those due to Leopoldt, Armitage-Frohlich, and Gras.

This is joint work with David S. Dummit and Richard Foote and with Ben Breen, Noam Elkies, and Ila Varma.

number theory

Audience: researchers in the topic

Chicago Number Theory Day 2020

Series comments: Description: Conference

A virtual number theory conference on Saturday, June 20, 2020 via Zoom. Please register on the conference website to receive the Zoom link.