Arithmetic of Bruhat-Tits Group Schemes over a Semi-Local Dedekind Ring
Bruhat-Tits group schemes are remarkable integral models of algebraic groups over non-archimedean henselian discretely valued fields, arising from affine buildings. They remain a subject of active research, highlighted by recent publications such as the book by Kaletha and Prasad. While this notion extends naturally to general semi-local Dedekind rings, injectivity questions "à la Grothendieck-Serre" was hardly addressed by Bruhat and Tits in the henselian case and remains virtually unexplored in the semi-local case. Existing results are limited: the case where the model is reductive corresponds to the Grothendieck-Serre conjecture over a semi-local ring, which admits a positive answer (Nisnevich, Guo). Furthermore, Bayer-Fluckiger and First questioned whether this injectivity extends to general Bruhat-Tits group schemes during their study of hereditary orders. Our main objective is to identify the obstruction to satisfying this injectivity "à la Grothendieck-Serre." We adopt Nisnevich's strategy: reducing the problem to the henselian case and resolving a decomposition problem. In this thesis, we first refine the cohomological study of Bruhat-Tits subgroups in the henselian case, complementing the Tokyo article of Bruhat and Tits. By developing a combinatorial understanding involving affine Tits indices, we show that the obstruction is trivial in the simply connected semisimple case. Moreover, we compute the obstruction exactly for quasi-split adjoint groups, identifying specific classes where injectivity fails. Finally, we transition to the semi-local case by addressing the decomposition problem via an innovative application of Prasad's theorem on the G(K)+ subgroup. Consequently, we obtain a simplified proof of the Grothendieck-Serre conjecture and establish the injectivity result for semisimple simply connected groups over semi-local Dedekind rings.
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