### Abstract

Let k be a field, let G be a reductive group, and let V be a linear representation of G. Let V//G = Spec(Sym^{*}(V^{*}))^{G} denote the geometric quotient and let π:V→V//G denotethequotientmap.Arithmeticinvarianttheorystudiesthe map on the level of k-rational points. In this article, which is a continuation of the results of our earlier paper “Arithmetic invariant theory”, we provide necessary and sufficient conditions for a rational element of V// G to lie in the image of, π assuming that generic stabilizers are abelian. We illustrate the various scenarios that can occur with some recent examples of arithmetic interest.

Original language | English (US) |
---|---|

Pages (from-to) | 139-171 |

Number of pages | 33 |

Journal | Progress in Mathematics |

Volume | 312 |

DOIs | |

State | Published - Jan 1 2015 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Keywords

- Galois cohomology
- Representation theory

## Fingerprint Dive into the research topics of 'Arithmetic invariant theory II: Pure inner forms and obstructions to the existence of orbits'. Together they form a unique fingerprint.

## Cite this

Bhargava, M., Gross, B. H., & Wang, X. (2015). Arithmetic invariant theory II: Pure inner forms and obstructions to the existence of orbits.

*Progress in Mathematics*,*312*, 139-171. https://doi.org/10.1007/978-3-319-23443-4_5