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Organizing Committee
Eastern Michigan University
Utrecht University
University of Wisconsin-Madison
University of Bath
Massachusetts Institute of Technology
Abstract
A conjecture of Malle predicts an asymptotic formula for the quantity of number fields with a given Galois group and whose discriminant is bounded by a growing parameter. This conjecture is foundational to the area of Arithmetic Statistics and has received considerable interest over the past 20 years, notably including Bhargava receiving the Fields Medal in part for his work on quartic and quintic field extensions.
Progress has been so rapid and varied, that there are now a host of cases known depending on a variety of group and number theoretic parameters. There are also numerous improvements to the original conjecture due to Bhargava, Wang, and Loughran--Santens that sharpen its predictions. The framework now consists of a range of mathematical tools, including permutation groups, embedding problems, group cohomology, algebraic stacks, and Brauer groups. Determining what follows from existing results for a particular Galois group, or even determining what the exact prediction should be, is a non-trivial endeavor.
The aims of the workshop are to overview the current state of the art in Malle's conjecture, to compile a living database regarding progress on the conjecture, and to develop algorithms to calculate the constants arising in Malle's conjecture. The first day will consist of talks. During the remaining time, participants will join focused working groups developing algorithms and inputting data into the L-functions and Modular Forms Database.
The deadline to apply for this workshop has been extended to February 16, 2026.
Progress has been so rapid and varied, that there are now a host of cases known depending on a variety of group and number theoretic parameters. There are also numerous improvements to the original conjecture due to Bhargava, Wang, and Loughran--Santens that sharpen its predictions. The framework now consists of a range of mathematical tools, including permutation groups, embedding problems, group cohomology, algebraic stacks, and Brauer groups. Determining what follows from existing results for a particular Galois group, or even determining what the exact prediction should be, is a non-trivial endeavor.
The aims of the workshop are to overview the current state of the art in Malle's conjecture, to compile a living database regarding progress on the conjecture, and to develop algorithms to calculate the constants arising in Malle's conjecture. The first day will consist of talks. During the remaining time, participants will join focused working groups developing algorithms and inputting data into the L-functions and Modular Forms Database.
The deadline to apply for this workshop has been extended to February 16, 2026.
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Workshop Schedule
Monday, June 08, 2026
8:30 - 8:50 AM EDT
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EDT
On b-constant in Malle's conjecture
11th Floor Lecture Hall
Speaker
Jiuya Wang, University of Georgia
Session Chair
Brandon Alberts, Eastern Michigan University
Abstract
In this talk we will give a summary on all known issues on $b$-constant in Malle's conjecture and its natural generalizations. We will explain the difference between global function fields and number fields, and propose a new refinement for Malle's conjecture over number fields.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
Number field counting and the Brauer group
11th Floor Lecture Hall
Speaker
Tim Santens, Cambridge University
Session Chair
Brandon Alberts, Eastern Michigan University
Abstract
Loughran and I have formulated a conjecture on the leading constant in the asymptotics of certain number field counting problems. A crucial new object in this conjecture is the Brauer group. I will explain what the Brauer group is and how it provides obstructions to the existence of extensions of number fields with certain local properties. Finally, I will describe the conjectural effect on the leading constant and how it can be computed.
11:30 AM - 12:15 PM EDT
Inductive Methods for Counting Number Fields
11th Floor Lecture Hall
Speaker
Melanie Wood, Harvard University
Session Chair
Brandon Alberts, Eastern Michigan University
Abstract
We describe an inductive method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle's Conjecture and counterexamples to Malle's Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group G. Our method relies on having asymptotic counts for certain T-extensions for some normal subgroup T of G, uniform bounds for the number of such T-extensions, and possibly weak bounds on the asymptotic number of G/T-extensions. However, we do not require that most T-extensions of a G/T-extension are G-extensions, i.e. we are not just counting wreath products. Our new results use T either abelian or S_m^3, though our framework is general. This talk is on joint work with Brandon Alebrts, Robert Lemke Oliver, and Jiuya Wang.
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:15 PM EDT
Bounds for ℓ-torsion in class groups
11th Floor Lecture Hall
Speaker
Stephanie Chan, University College London
Session Chair
Peter Koymans, Utrecht University
Abstract
We review some known pointwise and average bounds on the ℓ-torsion subgroup of class groups of number fields. Most current methods rely on the presence of many small split primes, an approach made explicit in the work of Ellenberg and Venkatesh. We will explain how the existence of these primes leads to non-trivial bounds for ℓ-torsion.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
A user's guide to bounding number fields
11th Floor Lecture Hall
Speaker
Robert Lemke Oliver, University of Wisconsin-Madison
Session Chair
Peter Koymans, Utrecht University
Abstract
There are many results that give upper bounds on the number of number fields of given degree and bounded discriminant, possibly with some extra structure (e.g. Galois group) or ordered in a different way. In this talk, we'll overview these results, how they fit together, and how to best think about obtaining upper bounds for a given family of number fields. We'll also aim to sell the problem of obtaining "inertial multiplicity bounds" for families of number fields. We expect this perspective will be useful both this workshop and more broadly.
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, June 09, 2026
9:00 - 9:45 AM EDT
Malle's conjecture in the LMFDB
11th Floor Lecture Hall
Speaker
David Roe, Massachusetts Institute of Technology
Session Chair
Robert Lemke Oliver, University of Wisconsin-Madison
Abstract
I will provide an introduction to the L-functions and modular forms database, focused on the improvements that we hope will come out of this week's workshop. In particular, I will provide a tour of the number field, p-adic field, abstract group and Galois group sections and discuss what information we would like to add. I will also sketch some recent improvements in the database of p-adic fields, and ask questions about how the recently added families of p-adic fields can connect to the counting problems arising in the Tamagawa measure part of the leading coefficient c(k, G, H) in Malle's conjecture.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
Counting number fields using multiple Dirichlet series
11th Floor Lecture Hall
Speaker
Alina Bucur, UCSD
Session Chair
Robert Lemke Oliver, University of Wisconsin-Madison
Abstract
We will discuss how to use multiple Dirichlet series to set up a generating function that helps us count number fields of fixed degree and Galois group by general invariants. The strategy works for certain semiconcentrated Galois groups, but it is limited by (sub)convexity bounds for the growth of L-functions inside the critical strip. This is joint work with Brandon Alberts.
11:15 AM - 12:30 PM EDT
Group Discussion/ Create Working Groups
Group Work - 11th Floor Lecture Hall
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:15 PM EDT
Group Work
Assigned Locations
3:15 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:00 PM EDT
Group Work
Assigned Locations
4:00 - 5:00 PM EDT
Report Out
Group Presentations - 11th Floor Lecture Hall
Wednesday, June 10, 2026
9:00 - 9:45 AM EDT
Computing unramified Brauer groups
11th Floor Lecture Hall
Speaker
Lucas Lagarde, Laboratoire Analyse, Géométrie et Applications Institut Galilée – Université Sorbonne Paris Nord
Session Chair
Daniel Loughran, University of Bath
Abstract
The unramified Brauer groups of classifying stacks of finite groups encode subtle arithmetic information (conjectural leading constants in Malle's conjecture, local-global analogues of the inverse Galois problem) and are highly difficult to compute in general. I will present explicit formulas for these groups, together with an effective algorithm for calculating them. Some families of examples illustrating the method will be given.
9:55 - 10:00 AM EDT
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 AM - 12:30 PM EDT
Group Work
Assigned Locations
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:15 PM EDT
Group Work
Assigned Locations
3:15 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:00 PM EDT
Group Work
Assigned Locations
4:00 - 5:00 PM EDT
Report Out
Group Presentations - 11th Floor Lecture Hall
Thursday, June 11, 2026
9:00 - 9:45 AM EDT
2-torsion ideal classes in pure cubic fields
11th Floor Lecture Hall
Speaker
Ari Shnidman, Temple University
Session Chair
David Roe, Massachusetts Institute of Technology
Abstract
I'll discuss recent work with Siad in which we compute the average number of 2-torsion ideal classes in pure cubic extensions over certain number fields. This has direct application to counting questions for certain types of S4/A4 quartic extensions. In the process of proving the result, we notice a phenomenon that leads us to predict an altered form of the Cohen-Lenstra heuristics for p-torsion classes in certain special families of number fields.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 AM - 12:30 PM EDT
Group Work
Assigned Locations
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:15 PM EDT
Group Work
Assigned Locations
3:15 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:00 PM EDT
Group Work
Assigned Locations
4:00 - 5:00 PM EDT
Report Out
Group Presentations - 11th Floor Lecture Hall
Friday, June 12, 2026
9:00 - 10:00 AM EDT
Group Work
Assigned Locations
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:30 AM EDT
Group Work
Assigned Locations
11:30 AM - 12:30 PM EDT
Report Out
Group Presentations - 11th Floor Lecture Hall
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