A Simple Proof of the Mean Value of $\left|K_2(\mathcal{O})\right|$ in Function 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.

6 pages, to appear in Comptes Rendus Mathematique

References in corpus (1)

• The Mean Value of $L(\frac{1}{2},\chi )$ in the Hyperelliptic Ensemble