On tamely ramified $\mathcal G$-bundles on curves

TL;DR

This paper studies the structure of tamely ramified group schemes and torsors over curves, establishing their relation to reductive group schemes over tame covers and developing a theory of local types.

Contribution

It introduces a framework connecting Bruhat-Tits group schemes and reductive groups over tame covers, including a theory of local types for equivariant torsors.

Findings

Group schemes can be expressed as invariants of reductive groups over tame covers.

A theory of local types for equivariant torsors is developed.

Relations between moduli stacks of torsors under different group schemes are established.

Abstract

We consider parahoric Bruhat-Tits group schemes over a smooth projective curve and torsors under them. If the characteristic of the ground field is either zero or positive but not too small and the generic fiber is absolutely simple and simply-connected, we show that such group schemes can be written as invariants of reductive group schemes over a tame cover of the curve. We relate the torsors under the Bruhat-Tits group scheme and torsors under the reductive group scheme over the cover which are equivariant for the action of the covering group. For this, we develop a theory of local types for such equivariant torsors. We also relate the moduli stacks of torsors under the Bruhat-Tits group scheme and equivariant torsors under the reductive group scheme over the cover. In an Appendix, B. Conrad provides a proof of the Hasse principle for adjoint groups over function fields with finite field of constants.