Arithmetic of Bruhat-Tits group schemes over semilocal dedekind rings

par Anis Zidani

jeudi 1 oct. 2026, 14:00 → 19:00 Europe/Paris

M7-411 (UMPA)

Let R be a DVR and G a reductive group over K = Frac(R). We say P is a Bruhat-Tits group scheme over R if, for every maximal ideal m of R, P is Bruhat-Tits (in the usual sense) over the completion of R at m.
In our setting, a Bruhat-Tits group scheme over a complete DVR can be a parahoric group scheme, the stabilizer of a point, or even the lft Néron model of a torus, or even more exotic group schemes.

The key question of the talk is to understand when the map H^1(R,P) --> H^1(K,G) is injective.
This question was originally posed by Bayer and First in their study of classical groups and hereditary orders.
The Grothendieck-Serre conjecture over R (proved by Nisnevich and Guo) is the special case where P is reductive over R.

We first lay the groundwork for studying the question, then show that the map is always injective when G is semisimple simply connected (for any P).
We also give some counterexamples where injectivity fails.
We also give a simplified proof of the Grothendieck-Serre conjecture over R, a proof that fits more within a building approach.