On the rational K_2 of a curve of GL_2 type over a global field of positive characteristic

摘要

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 "GL2 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 Delliptic sheaves of rank 2.