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 K1(X) injects into K1(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 G1 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.
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