Arithmetic invariant theory

Manjul Bhargava, Benedict H. Gross

Research output: Chapter in Book/Report/Conference proceedingChapter

18 Scopus citations


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* (Vv)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 = SO2n+1.

Original languageEnglish (US)
Title of host publicationProgress in Mathematics
PublisherSpringer Basel
Number of pages22
StatePublished - 2014

Publication series

NameProgress in Mathematics
ISSN (Print)0743-1643
ISSN (Electronic)2296-505X

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology


  • Hyperelliptic curves
  • Invariant theory


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