Asymptotic growth of orders in a fixed number field via subrings in $\mathbb{Z}^n$

Abstract: Let $K$ be a number field of degree $n$ and $\mathcal{O}_K$ be its ring of integers. An order in $\mathcal{O}_K$ is a finite index subring that contains the identity. A major open question in arithmetic statistics asks for the asymptotic growth of orders in $K$. In this talk, we will give the best known lower bound for this asymptotic growth. The main strategy is to relate orders in $\mathcal{O}_K$ to subrings in $\mathbb{Z}^n$ via zeta functions. Along the way, we will give lower bounds for the asymptotic growth of subrings in $\mathbb{Z}^n$ and for the number of index $p^e$ subrings in $\mathbb{Z}^n$. We will also discuss analytic properties of these zeta functions.

number theory

Audience: researchers in the topic

Comments: There will be a pretalk at 1:30 Pacific time.

UCSD number theory seminar

Series comments: Most talks are preceded by a pre-talk for graduate students and postdocs. The pre-talks start 40 minutes prior to the posted time (usually at 1:20pm Pacific) and last about 30 minutes.