Nilpotent counting problems in arithmetic statistics

Nilpotent counting problems in arithmetic statistics

This workshop, sponsored by AIM and the NSF, will be devoted to nilpotent counting problems in arithmetic statistics. These include the closely related problems of counting number field extensions with nilpotent Galois group and bounded discriminant and the distribution of $p$-torsion in the class group for extensions of degree divisible by $p$. Significant recent progress has been made in this area producing a number of significant, but isolated, new results using a variety of new independent techniques.

The aim of this workshop is to bring these methods together, and investigate the potential for combining techniques to prove stronger results. The workshop also aims to study minimal, nontrivial working examples of these methods in new settings both as a means to further develop these tools and to increase access to the methods.

The main topics for the workshop are counting number fields with nilpotent Galois group and the distribution of the $p$-torsion of the class group for extensions of degree divisible by $p$, with methods including

• Smith's methods for proving the distribution of $2^\infty$-torsion in class groups of quadratic extensions. For example, the first moment of 8-torsion of class groups of quadratic fields relates to $D_8$-extensions of $\mathbb{Q}$, for which Malle's conjecture has yet to be verified.
• Solutions to central embedding problems, as employed by Koyman and Pagano to parametrize nilpotent extensions. Explicit presentations of Galois groups, including those proven by Liu.
• Multiple Dirichlet series methods for producing an analytic continuation of a generating series, as utilized by Altug, Shankar, Varma, and Wilson.
• Galois cohomology and the study of twisted counting problems studied by Alberts and O'Dorney.
• Geometric techniques for function field analogs and the insight they can give into the number field case, as in work by Liu, Wood, and Zureick-Brown.

This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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