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

Alternatively have a look at the program.

## Distributions of unramified extensions of global fields

We give conjectures on the distribution of the Galois groups of the maximal unramified extensions of Galois $\Gamma$ number fields or function

fields for any finite group $\Gamma$ (for the part of the Galois group prime to the order of $\Gamma$ and the order of roots of unity in the base

field). We explain some results about these Galois groups that motivate us to build certain random groups whose distributions appear in our

conjectures. We give theorems in the function field case (as the size of the finite field goes to infinity) that support these new

## Geometric arithmetic statistics

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,

## Redei's reciprocity law

I explain a corrected 21st century reformulation and give the first complete proof of a reciprocity law from 1939 that is due to Redei, and that played a central role in Smith's 2016 preprint on quadratic 2-class groups.

## Class groups, Selmer groups and Cassels--Tate pairings

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.

## $2^k$-Selmer groups and Goldfeld's conjecture, I

Take $E$ to be an elliptic curve over a number field whose four torsion obeys certain technical conditions. In this talk, we will outline a proof that $100\%$ of the quadratic twists of $E$ have rank at most one. To do this, we will find the distribution of $2^k$-Selmer ranks in this family for every $k > 1$. Using this framework, we will also find the distribution of the $2^k$-class ranks of the imaginary quadratic fields for all $k > 1$.

## The 2-torsion subgroups of the class groups of cubic fields

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.

## Justifying Random Models of Nonabelian Class Groups

Recent works by Boston, Wood, and others suggest a balanced presentation random model for 'nonabelian class groups'. We show that over function fields, these class groups (etale fundamental groups) are indeed balanced.

## $2^k$-Selmer groups and Goldfeld's conjecture, II

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 $E^d$.

## Arithmetic statistics via graded Lie algebras

I will talk about recent work with Jack Thorne in which we find the average size of the 3-Selmer group for a family of genus-2 curves by analyzing a graded Lie algebra of type $E_8$. I will focus on the role representation theory plays in our proofs.

## $2^k$-Selmer groups and Goldfeld's conjecture, III

In this combinatorial talk, we will prove the partial information found in the second talk is enough to prove the claimed distribution for $2^k$-Selmer groups over grids. We will focus primarily on the case of $4$-Selmer groups.

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