A Simple Proof of the Mean Value of $\left|K_{2}(\mathcal{O})\right|$ in Function Fields

TL;DR

This paper provides a straightforward proof for the average size of the K_2 groups of certain function field rings, using character sums and the Riemann hypothesis for curves over finite fields.

Contribution

It offers a simple, elegant proof of the mean value of |K_2(O_D)| in function fields, leveraging character sum estimates and the Riemann hypothesis.

Findings

Average size of |K_2(O_D)| computed explicitly

Proof relies on character sum estimates

Utilizes the Riemann hypothesis for curves over finite fields

Abstract

Let $F$ be a finite field of odd cardinality $q$, $A=F[T]$ the polynomial ring over $F$, $k=F(T)$ the rational function field over $F$ and $\mathcal{H}$ the set of square-free monic polynomials in $A$ of degree odd. If $D\in\mathcal{H}$, we denote by $\mathcal{O}_{D}$ the integral closure of $A$ in $k(\sqrt{D})$. In this note we give a simple proof for the average value of the size of the groups $K_{2}(\mathcal{O}_{D})$ as $D$ varies over the ensemble $\mathcal{H}$ and $q$ is kept fixed. The proof is based on character sums estimates and in the use of the Riemann hypothesis for curves over finite fields.