## Number Theory - Anup Dixit (Queen's University)

### Tuesday, July 24th, 2018

**Time:** 2:30-3:20p.m. **Place:** Jeffery Hall 422

**Speaker:** Anup Dixit (Queen's University)

**Title:** ON THE GENERALIZED BRAUER-SIEGEL THEOREM.

**Abstract:** For a number field $K/ \mathbb{Q}$, the class number $h_K$ captures how far the ring of integers of $K$ is from being a PID. The study of class numbers is an important theme in number theory. In order to understand how the class number varies upon varying the number field, Siegel showed that the class number times the regulator tends to infinity in any family of quadratic number fields. Brauer extended this result to family of Galois extensions over $\mathbb{Q}$. This is the Brauer-Siegel theorem. Recently, Tsfasman and Vladut conjectured a Brauer-Siegel statement for an asymptotically exact family of number fields. In this talk, we prove the classical Brauer-Siegel and the generalized version in several unknown cases. We also discuss some effective versions of Brauer-Siegel in the classical setting.