On the rational K(2) of a curve of GL(2) type over a global field of positive characteristic

Abstract

. If X is an integral model of a smooth curve X over a global field k , there is a localization sequence comparing the K -theory of X and X . We show that K 1 ( X ) injects into K 1 ( X ) rationally, by showing that the previous boundary map in the localization sequence is rationally a surjection, for X of “GL 2 type” and k of positive characteristic not 2. Examples are given to show that the relative G 1 term can have large rank. Examples of such curves include non-isotrivial elliptic curves, Drinfeld modular curves, and the moduli of D -elliptic sheaves of rank 2.