If script 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 script X and X. We show that K1(script 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 script D-elliptic sheaves of rank 2.
- Parshin conjecture
- function fields