Sameera Vemulapalli: Counting Low Degree Number Fields with Almost Prescribed Successive Minima

26th January, 2023

The successive minima of an order in a degree n number field are n real numbers encoding information about the Euclidean structure of the order. How many orders in degree n number fields are there with almost prescribed successive minima, fixed Galois group, and bounded discriminant? In this talk, I will address this question for n = 3,4,5. The answers, appropriately interpreted, turn out to be piecewise linear functions on certain convex bodies. If time permits, I will also discuss function field analogues of this problem.

Melanie Matchett Wood: A visit to 3-manifolds in the quest to understand random Galois groups

13th October, 2022

Cohen, Lenstra, and Martinet gave conjectural distributions for the class group of a random number field. Since the class group is the Galois group of the maximum abelian unramified extension, a natural generalization would be to give a conjecture for the distribution of the Galois group of the maximal unramified extension. Previous work (joint with Liu and Zurieck-Brown) produced a plausible conjecture for the part of this Galois group relatively prime to the number of roots of unity in the base field. There is a deep analogy between number fields and 3-manifolds. Thus, an analogous question would be to describe the distribution of the profinite completion of the fundamental group of a random 3-manifold. In this talk, I will explain how Will Sawin and I answered this question for a model of random 3-manifolds defined by Dunfield and Thurston, and how the techniques we used should allow us, in future work, to prove large q limit theorems in the function field analogue and give a general conjecture in the number field case, taking into account roots of unity in the base field. This is part two of a series of two talks on joint work, some in progress, with Will Sawin. Both talks should be understandable on their own.

Will Sawin: New Cohen-Lenstra heuristics by constructing measures from moments

6th October, 2022

The Cohen-Lenstra heuristics give predictions for the distribution of the class groups of a random quadratic number field. Cohen and Martinet generalized them to predict the distribution of the class groups of random extensions of a fixed base field, but Malle pointed out that these predictions have errors arising from the roots of unity in the base field. We give amended predictions that account for the influence of roots of unity.

Our predictions are based on a result which produces a formula for the distribution of a random finite abelian group given its moments, i.e., the expected number of surjections onto a fixed group. This result is very general and we expect it to have further applications in arithmetic statistics.

This is part one of a series of two talks on joint work, some in progress, with Melanie Matchett Wood. Both talks should be understandable on their own.

Manjul Bharğava: Galois groups of random integer polynomials

21st April, 2022

Of the (2H+1)n monic integer polynomials f(x)=xn+a1xn−1+⋯+an with max{|a1|,…,|an|}≤H, how many have associated Galois group that is not the full symmetric group Sn? There are clearly ≫Hn−1 such polynomials, as may be obtained by setting an=0. In 1936, van der Waerden conjectured that O(Hn−1) should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees n≤4, due to work of van der Waerden and Chow and Dietmann. In this talk, we will describe a proof of van der Waerden's Conjecture for all degrees n.

Robert Lemke Oliver: The average size of 3-torsion in class groups of 2-extensions

7th April, 2022

We determine the average size of the 3-torsion in class groups of G-extensions of a number field when G is any transitive 2-group containing a transposition, for example, D4. It follows from the Cohen-Lenstra-Martinet heuristics that the average size of the p-torsion in class groups of G-extensions of a number field is conjecturally finite for any G and most p. Previously this conjecture had only been proved in the cases of G=S2 with p=3 and G=S3 with p=2. We also show that the average 3-torsion in a certain relative class group for these G-extensions is as predicted by Cohen and Martinet, proving new cases of the Cohen-Lenstra-Martinet heuristics. Our new method also works for many other permutation groups G that are not 2-groups.

Ashwin Iyengar: Local (ℓ=p) Galois Deformation Rings

10th February, 2022

I will present joint work with V. Paškūnas and G. Böckle concerning deformation rings for mod p Galois representations of p-adic local fields. After giving a short introduction to the subject, I will explain our main result which says that framed local deformation rings are complete intersections of the "expected dimension", and which gives a classification of their irreducible components in terms of a determinant map. I will explain some of the ingredients that go into our proof, which involves work on pseudo-deformations by Böckle-Juschka, and moduli spaces of representations with fixed pseudo-character. If time permits I will discuss an application to density of crystalline points in deformation spaces.

Jacob Tsimerman: Abelian varieties not isogenous to Jacobians

1st December, 2021

Katz and Oort raised the following question: Given an algebraically closed field k, and a positive integer g>3, does there exist an abelian variety over k not isogenous to a Jacobian over k? There has been much progress on this question, with several proofs now existing over ℚ. We discuss recent work with Ananth Shankar, answering this question in the affirmative over 𝔽q(T). Our method introduces new types of local obstructions, and can be used to give another proof over ℚ.

Alexandre Grishkov: From 12th Hilbert problem to Carlitz-Drinfeld-Anderson modules

11th November, 2021

The well-known Kronecker-Weber theorem affirms that every finite abelian extension of the field ℚ of rational numbers belongs to some cyclotomic extension ℚ(t|tn=1). In his 12th problem D.Hilbert asked how to generalize this theorem for other global fields. In this talk, we give the exposition of the actual state of this problem together with the connection with Carlitz-Drinfeld-Anderson modules. Recall that Anderson module M is a (left) module over non-commutative ring R=Cp[T,τ], Tτ=τT, τa=ap τ, where Cp is a some field of characteristic p greater than 0, such that M is free finite generated over subrings Cp[T] and Cp[τ].

Artane Jeremie Siad: Monogenic fields with odd class number

4th November, 2021

In this talk, we prove an upper bound on the average number of 2-torsion elements in the class group of monogenised fields of any degree n≥3 and, conditional on a widely expected tail estimate, compute this average exactly. As an application, we show the existence of infinitely many number fields with odd class number in almost every even degree and signature. Time permitting, we will also discuss extensions of these results to orders (joint with Shankar, Swaminathan and Varma) and the relative setting.