Manjul Bhargava

Arithmetic invariant theory

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

Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves

Using maximal isotropic submodules in a quadratic module over ℤ_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of cofinite type ℤ_p-modules, and then conjecture that as E varies over elliptic curves over a fixed global field k, the distribution of 0 → E(k)⊗ ℚ_p/ ℤ_p → Selp∞ E → Ш[p^∞ ] → 0 is that one. We show that this single conjecture would explain many of the known theorems and conjectures on ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves. We also prove the existence of a discrete probability distribution on the set of isomorphism classes of finite abelian p-groups equipped with a nondegenerate alternating pairing, defined in terms of the cokernel of a random alternating matrix over ℤ_p, and we prove that the two probability distributions are compatible with each other and with Delaunay’s predicted distribution for Ш. Finally, we prove new theorems on the fppf cohomology of elliptic curves in order to give further evidence for our conjecture.

Gauss Composition and Generalizations

We discuss several higher analogues of Gauss composition and consider their potential algorithmic applications.

The density of discriminants of ₃-sextic number fields

. We prove an asymptotic formula for the number of sextic number fields with Galois group S 3 and absolute discriminant < X. In addition, we give an interpretation of the constant in the formula in terms of the asymptotic densities of given local completions among these sextic fields. Our proof gives analogous results when we count S 3 -sextic extensions of any number field, and also when finitely many local completions have been specified for the sextic extensions.

On the mean number of

Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2-torsion elements in the class groups and narrow class groups of these cubic fields when ordered by their absolute discriminants. For an order

2

\cal O

-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields

in a cubic field, we study the three groups:

The proportion of plane cubic curves over ℚ that everywhere locally have a point

\rm Cl_2(\cal O)

Arithmetic invariant theory II: Pure inner forms and obstructions to the existence of orbits

, the group of ideal classes of

The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields

\cal O