## Arithmetic Geometry and Number Theory RTG Seminar: Odd degree number fields with odd class number

Seminar | October 16 | 3:10-5 p.m. | 891 Evans Hall

Wei Ho, University of Michigan

Department of Mathematics

For any fixed odd integer $n \geq 3$, we study the 2-torsion of the ideal class groups of certain families of degree $n$ number fields. We show that (up to a tail estimate) the average size of the 2-torsion in these families matches the predictions given by the Cohen-Lenstra-Martinet-Malle heuristics, which predict the distribution of class groups of number fields. As a consequence, we find that for any odd $n\geq 3$, there exist infinitely many number fields of degree $n$ and associated Galois group $S_n$ whose class number is odd. This talk is based on joint work with Arul Shankar and Ila Varma.

Seminar Format: The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.

yxy@berkeley.edu