an:06302771
Zbl 1317.14101
Gille, Philippe; Moret-Bailly, Laurent
Algebraic actions of arithmetic groups
FR
Skorobogatov, Alexei N. (ed.), Torsors, ??tale homotopy and applications to rational points. Lecture notes of mini-courses presented at the workshop ``Torsors: theory and applications'', Edinburgh, UK, January 10--14, 2011 and at the study group organised in Imperial College, London, UK in autumn 2010. Cambridge: Cambridge University Press (ISBN 978-1-107-61612-7/pbk; 978-1-139-52535-0/ebook). London Mathematical Society Lecture Note Series 405, 231-249 (2013).
2013
a
14L15 14G25 14L05
arithmetic group; algebraic group; torsor
In this article, the authors establish various finiteness results concerning the \(H^1_{\mathrm{fppf}}\) of affine groups in \(p\)-adic and global characteristic-zero settings, using the modern theory of group schemes and torsors. Based on these results, they obtain the following generalization of a finiteness theorem of Platonov: Let \(S\) be a finite set of finite places of a number field \(F\), and let \(A_S\) be the ring of \(S\)-integers in \(F\). Consider a group \(A_S\)-scheme \(G\) and a flat \(A_S\)-scheme \(X\) of finite type equipped with a left \(G\)-action. Let \(Z_0 \subset X\) be a closed \(A_S\)-subscheme which is flat over \(A_S\), and let \(\mathrm{loc}(Z_0)\) be the set of closed subschemes \(Z \subset X\) which are \(G(\overline{A_v})\)-translates of \(Z\) at each finite place \(v \notin S\), where \(\overline{A_v}\) stands for the ring of integers in \(\overline{F_v}\). Then \(G(A_S) \backslash \mathrm{loc}(Z_0)\) is finite.
The geometric, i.e. equal characteristic \(p > 0\) case is also discussed, and the finiteness theorem alluded to above requires stronger conditions.
For the entire collection see [Zbl 1277.14003].
Wen-Wei Li (Beijing)