### 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.

Original language | English |
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Pages (from-to) | 313-342 |

Number of pages | 30 |

Journal | Journal of K-Theory |

Volume | 14 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 Jun 30 |

### Keywords

- K-theory
- Parshin conjecture
- function fields

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

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## Cite this

Chida, M., Kondo, S., & Yamauchi, T. (2014). On the rational K

_{2}of a curve of GL_{2}type over a global field of positive characteristic.*Journal of K-Theory*,*14*(2), 313-342. https://doi.org/10.1017/is014006024jkt272