##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Kelly Isham

#### University of California Irvine

## 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.

Host: Kiran Kedlaya

### June 3, 2021

### 2:00 PM

### Location: See https://www.math.ucsd.edu/\~{}nts/

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