# 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

## 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.

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## Families of elliptic curves ordered by conductor

Conjectures on the statistics of elliptic curves are generally formulated with the assumption that the curves in question are ordered by their conductors. However, when proving results on the

## 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.

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## 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.

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