List of Titles and Abstracts

Tim Browning: Solubility of a resultant equation

Given an odd positive integer n, how likely is it for an integral quadratic polynomial to have resultant ±1 with some integral polynomial of degree n?  Aaron Landesman’s PhD thesis describes a connection between this question and averages of n-torsion of the class group of quadratic number fields. I’ll discuss both problems and show how the large sieve can be used to address them.  This is joint work with Stephanie Chan.

Stephanie Chan: Pointwise bounds for 3-torsion

For $\ell$ an odd prime number and $d$ a squarefree integer, a central question in arithmetic statistics is to give pointwise bounds for the size of the $\ell$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. This is in general a difficult problem, and unconditional pointwise bounds are only available for $\ell = 3$ due to work of Pierce, Helfgott—Venkatesh and Ellenberg—Venkatesh. The current record is $h_3(d) \ll_\epsilon d^{1/3 + \epsilon}$ due to Ellenberg—Venkatesh. We will discuss how to improve this to $h_3(d) \ll d^{0.32}$. This is joint work with Peter Koymans.

Jordan Ellenberg: New developments in arithmetic statistics over function fields

I will give a survey of the developing dictionary between representations of the braid group and arithmetic-statistical problems over function fields, including recent joint results with Mark Shusterman about class groups of quadratic extensions with prime discriminant.

Hendrik Lenstra: A cohomological invariant of wild extensions of local fields

Any finite extension of a local field can be split up in two pieces: a tame piece that is well-understood, and a totally wild piece that is somewhat mysterious. In the lecture we shall, to any finite separable field extension of a local field that is totally wild, attach an invariant that belongs to a certain cohomology group associated with the base field. The construction of this element makes use of a group-theoretic map related to the transfer map. For many local fields that one encounters in practice, the cohomology group is finite and has a known structure, and one may wonder how the invariant varies over the group as the extension field varies over all possibilities.  (Joint work with Matthijs Buise, Leiden.)

Yuan Liu: The imaginary case of the Nonabelian Cohen—Lenstra heuristics

The Cohen-Lenstra heuristics are about the distribution of the p-part of class group of quadratic number fields for odd prime p, and the nonabelian Cohen—Lenstra problem study the distribution of the Galois groups of the maximal unramified extensions of families of global fields. In my previous work joint with Wood and Zureick-Brown, we gave conjectures for the nonabelian Cohen—Lenstra problem for totally real number fields with a fixed Galois group over the rational numbers, and proved the function field analog of our conjecture. In this talk, we will discuss the obstacles to generalizing Liu—Wood—Zureick-Brown to the imaginary fields, and then show the solutions to overcome those difficulties. This talk is based on the joint work with Ken Willyard.  

Tim Santens: The leading constant in Malle's conjecture

Let G be a finite permutation group, Malle has put forward a conjecture on the number of G-extensions of a number field of bounded discriminant. There exists a superficially similar conjecture by Manin and his collaborators on the number of points of bounded height on varieties. In this talk I will discuss recent efforts to interpret Malle's conjecture as a form of Manin's conjecture for the stack BG. Based on this analogy Loughran and I have given a conjectural interpretation of the leading constant in Malle's conjecture.

Alex Smith: Tamagawa ratios and arithmetic statistics

Given a finite Galois module M over a number field, the ratio of the size of the Selmer group of M to the size of the Selmer group of the dual module M*(1) can be easily calculated even when the Selmer groups themselves cannot. This ratio is known as a Tamagawa ratio, and unusual behavior in the Selmer groups in a given family of Galois modules can often be explained in terms of the behavior of the Tamagawa ratio.

In this talk, we advance a conjecture that, in certain geometric families of elliptic curves over Q, the average size of the p-Selmer groups is unbounded if and only if there is some geometric p-isogeny such that the average size of its associated Tamagawa ratios is also unbounded. We show that this conjecture is true in all families where the Galois structure of the p-torsion is static throughout the family.

We also apply this framework to class groups. Given a G-extension K/F of number fields and a prime p dividing |G|, we give a conjectural decomposition of the p-part of the class group of K into a formulaic part and a part whose average size in the family of G extensions is bounded.

This work is joint in part with Peter Koymans and in part with Ken Willyard.

Ashvin Swaminathan: Second moments for 2-class groups in families of cubic fields

We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the 2-class group is at most 3 (resp., at most 6). In the totally real case, we further prove that the second moment of the size of the narrow 2-class group is at most 9. This result gives further evidence in support of the general observation, first made in work of Bhargava--Hanke--Shankar and recently formalized into a set of heuristics in work of Siad--Venkatesh, that monogenicity has an altering effect on class group distributions. We also present ongoing work in progress proving analogous theorems for families of non-monogenic cubic fields.

This is joint work with Manjul Bhargava and Arul Shankar.

Jiuya Wang: Dominant Galois groups for large degree number fields

Hilbert irreducibility theorem implies that a random degree n polynomial has full symmetric group as Galois group with probability 1. We investigate the parallel question for degree n number fields when ordered by discriminant. In degree less than six, it has been proved that full symmetric group is the dominant Galois group. We explore the situation for large degree extension based on Bhargava's heuristics in Malle's Conjecture. Our result shows that unlike small degree examples or random polynomial questions, in large composite degrees, symmetric group arises with a probability approaching zero when ordered by discriminant. We will give further discussion on those Galois groups that are dominant. 

Melanie Wood: Non-abelian Cohen-Lenstra-Martinet in the presence of roots of unity

We will explain function field results and number theoretic conjectures (and some results) about the distribution of the Galois groups of the maximal unramified extension of G-extensions of a fixed number field, in particular on the part prime to |G| but not necessarily prime to the number of roots of unity in the base field.  We explain the additional structure one needs to put on these Galois groups to get nice distributional results.  We use our recent work on determining a distribution from its moments along with analysis of components of Hurwitz spaces to find the desired distributions.  We prove non-existence results in the number theoretic setting when our conjectures say that certain groups appear with probability zero.  This talk is on joint work with Will Sawin.