### Abstract

Let k be a field, let G be a reductive algebraic group over k, and let V be a linear representation of G. Geometric invariant theory involves the study of the k-algebra of G-invariant polynomials on V , and the relation between these invariants and the G-orbits on V , usually under the hypothesis that the base field k is algebraically closed. In favorable cases, one can determine the geometric quotient V//G = Spec(Sym* (V^{v})G) and can identify certain fibers of the morphism V → V//G with certain G-orbits on V. In this paper we study the analogous problem when k is not algebraically closed. The additional complexity that arises in the orbit picture in this scenario is what we refer to as arithmetic invariant theory. We illustrate some of the issues that arise by considering the regular semisimple orbits—i.e., the closed orbits whose stabilizers have minimal dimension—in three arithmetically rich representations of the split odd special orthogonal group G = SO_{2n+1}.

Original language | English (US) |
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Title of host publication | Progress in Mathematics |

Publisher | Springer Basel |

Pages | 33-54 |

Number of pages | 22 |

DOIs | |

State | Published - Jan 1 2014 |

### Publication series

Name | Progress in Mathematics |
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Volume | 257 |

ISSN (Print) | 0743-1643 |

ISSN (Electronic) | 2296-505X |

### All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Keywords

- Hyperelliptic curves
- Invariant theory

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## Cite this

*Progress in Mathematics*(pp. 33-54). (Progress in Mathematics; Vol. 257). Springer Basel. https://doi.org/10.1007/978-1-4939-1590-3_3