2012/06/21 by Manjul Bhargava, Benedict H. Gross, Bhargava, Manjul +1 · 1 citation
Mathematics · #11E72 #14L24 #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1206.4774
openalex publication_date 2012/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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^*))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 semi-simple 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.