Remark on a Simple Proof of the Mean Value of $K_2(\mathcal{O})$ in Function Fields

TL;DR

This paper provides a simple proof for the average size of the $K_2$ group associated with certain quadratic extensions over function fields, extending understanding of algebraic $K$-theory in this setting.

Contribution

The paper introduces a straightforward proof for the mean value of $K_2( ext{ring of integers})$ in quadratic function field extensions, clarifying previous results.

Findings

Computed the average size of $K_2$ groups for quadratic extensions

Provided a simplified proof method for mean value calculations

Enhanced understanding of algebraic $K$-theory in function fields

Abstract

Let $\mathbb{F}_q$ denote a finite field of odd cardinality $q$, $\mathbb{A}=\mathbb{F}_q[T]$ the polynomial ring over $\mathbb{F}_q$ and $k=\mathbb{F}_q(T)$ the rational function field over $\mathbb{F}_q$. In this paper, we compute the average value of the size of the group $K_2(\mathcal{O}_{\gamma D})$, where $\mathcal{O}_{\gamma D}$ denotes the integral closure of $\mathbb{A}$ in $k(\sqrt{\gamma D})$, $D$ is a monic, square-free polynomial of even degree and $\gamma$ is a fixed generator of $\mathbb{F}_q^*$.