Arithmetic invariant theory II: pure inner forms and obstructions to the existence of orbits. (English) Zbl 1377.11046

Bhargava, Manjul; Gross, Benedict H.; Wang, Xiaoheng

Nevins, Monica (ed.) et al., Representations of reductive groups. In honor of the 60th birthday of David A. Vogan, Jr. Proceedings of the conference, MIT, Cambridge, MA, USA, May 19–23, 2014. Cham: Birkhäuser/Springer (ISBN 978-3-319-23442-7/hbk; 978-3-319-23443-4/ebook). Progress in Mathematics 312, 139-171 (2015).

Summary: Let \(k\) be a field, let \(G\) be a reductive group, and let \(V\) be a linear representation of \(G\). Let \(V/\!\!/G =\mathop{\mathrm{Spec}}\nolimits (\mathrm{Sym}^\ast(V^\ast))^G\) denote the geometric quotient and let \(\pi: V V/\!/G\) denote the quotient map. Arithmetic invariant theory studies the map \(\pi\) on the level of \(k\)-rational points. In this article, which is a continuation of the results of our earlier paper “Arithmetic invariant theory” [ibid. 257, 33–54 (2014; Zbl 1377.11045)], we provide necessary and sufficient conditions for a rational element of \(V/\!\!/G\) to lie in the image of \(\pi\), assuming that generic stabilizers are abelian. We illustrate the various scenarios that can occur with some recent examples of arithmetic interest.
For the entire collection see [Zbl 1336.22001].

MSC:

11E72	Galois cohomology of linear algebraic groups
14L24	Geometric invariant theory

Keywords:

representation theory; Galois cohomology

Citations:

Zbl 1377.11045

References: