Average value of \(| K_ 2({\mathcal O})|\) in function fields
type: Item
instance of: scholarly article
title: Average value of \(| K_ 2({\mathcal O})|\) in function fields
zbMATH Open document ID: 0828.11069
published in: Finite Fields and their Applications
publication date: 1996-01-07T00:00:00Z
review text: Let \(F\) be a finite field with \(q\) elements, \(A = F[T]\) the polynomial ring over \(F\) and \(k = F(T)\). For any square-free polynomial \(m\) in \(A\) let \({\mathcal O}_m\) denote the integral closure of \(A\) in \(k(\sqrt m)\). The author obtains a formula for the average value of the size of the \(K\)-groups \(K_2 ({\mathcal O}_m)\), where \(m\) varies over all square- free polynomials of a fixed degree \(M\). The proof uses the relation between the size of \(K_2 ({\mathcal O}_m)\) and the value of the zeta- function of \({\mathcal O}_m\) at \(- 1\) given by the Birch-Tate conjecture. In the situation considered this conjecture is known to be true by a result of Tate.
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 759430
zbMATH Keywords: size of \(K_ 2 ({\mathcal O}_ m)\)
zbMATH Keywords: global field
zbMATH Keywords: finite field
zbMATH Keywords: zeta- function
zbMATH Keywords: Birch-Tate conjecture
author: Michael Rosen
reviewed by: Manfred Kolster
MaRDI profile type: Publication
full work available at URL: ffta.1995.1019
OpenAlex ID: W2050790424
DOI: 10.1006/FFTA.1995.1019
description: scientific article; zbMATH DE number 759430