Average value of \(| K_ 2({\mathcal O})|\) in function fields (Q1891288)

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.