Arithmetic invariant theory

By Manjul Bhargava, B. Gross. 2012

54 citations recorded by the source corpus.

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 \(G = \mathrm{SO}_{2n+1}\).

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c6c72cc4960ae070c0ed871241c898d651b3e1d6

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point (2012, background): “(k) 6= 2 . In this section and the next, we consider the orbits of the group G(k) = SO(W)(k) on the representation V(k) = V ⊗ k. A discussion of these orbits can also be found in our expository paper [4]. Since the group G(k)acts on the operators T in V(k)by conjugationT → gTg−1, the characteristic ”
Most hyperelliptic curves over Q have no rational points (2013, methodology): “ ⊗ Sym 2Z n having an irreducible invariant binary n-ic form of bounded height. The counting methods used build on those in [3], as well as those in our joint work with Shankar [9], Ho [8], and Gross [5], although as mentioned earlier the current scenario also has some new features in the cusps that”
Geometric representations of graded and rational Cherednik algebras (2014, background): “es that the intersection of with these lines is empty. The Jacobian J( ) of is an elliptic curve and ˘=Pic1( ) is a torsor under J( ). Since S acts freely on , it can only acts via translation by J( )[2], and we have a natural isomorphism S ˘=J( )[2] ˘=J(C )[2]. In particular, S acts trivially on H(”