Arithmetic invariant theory

  • Manjul BhargavaEmail author
  • Benedict H. Gross
Part of the Progress in Mathematics book series (PM, volume 257)


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 \(V /\!/G = \mathrm{Spec}(\mathrm{Sym}^{{\ast}}(V ^{\vee })^{G})\) and can identify certain fibers of the morphism \(V \rightarrow V/\!/G\) 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 \(G = \mathrm{SO}_{2n+1}\).


Invariant theory Hyperelliptic curves 

Mathematics Subject Classification 2010:

11E72 14L24 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics Princeton UniversityPrincetonUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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