Abstracts for Conference on "Enumerative Arithmetic and the Cohen–Lenstra Heuristics", June 3 - 7, 2019

We will explain the general philosophy by which one can relate questions in arithmetic statistics over Q to analogous questions over function fields of curves over $F_q$, and from there to questions about the topology of moduli spaces of complex algebraic varieties and morphisms. Our primary example will be the result of Ellenberg-Venkatesh-Westerland on geometric Cohen-Lenstra and the cohomology of Hurwitz spaces, but we will also try to indicate what this program might look like in the context of variation of Selmer groups,

In this largely expository talk we will give an introduction to certain Selmer groups associated to (compatible systems of) finite Galois modules, and describe an analogue of the Cassels--Tate pairing in this setting, due to Flach. We will then discuss the effect of twisting these modules by quadratic characters and show how class groups of quadratic fields, and Selmer groups of quadratic twist families of elliptic curves, naturally fit into this framework. If time permits we will discuss additional objects of arithmetic interest which can be understood from this viewpoint.

The Cohen–Lenstra–Martinet conjectures have been verified in only two cases. Davenport–Heilbronn compute the average size of the 3-torsion subgroups in the class group of quadratic fields and Bhargava computes the average size of the 2-torsion subgroups in the class groups of cubic fields. The values computed in the above two results are remarkably stable. For example, they do not change if one instead averages over the family of quadratic or cubic fields satisfying any finite set of splitting conditions.

Choose some positive $k$ and a rational elliptic curve $E$, and choose $k$ pairs of primes $(p_i, p'_i)$. Take $d_0$ to be $p_1 p_2 \dots p_k$, and consider the family of $d$ given by replacing $p_i$ with $p'_i$ for some set of $i$. Under special circumstances, we show that $2^k$-Selmer elements of a twist $E^{d_0}$ can be constructed from $2^k$-Selmer elements of the remaining twists $E^d$.