# Number Theory Seminar

This seminar is held on Tuesdays at 1pm in 105 Fenton Hall unless otherwise stated.

### Winter Quarter, 2019

- January 15,
**Maria Fox** (Boston College)

The GL(4) Rapoport-Zink Space
**Abstract**: The GL(2n) Rapoport-Zink space is a moduli space of supersingular p-divisible groups of dimension n and height 2n, with a quasi-isogeny to a fixed basepoint. After the GL(2) Rapoport-Zink space, which is zero-dimensional, the GL(4) Rapoport-Zink space has the most fundamental moduli description, yet relatively little of its specific geometry has been explored. In this talk, I will give a description of the geometry of the GL(4) Rapoport-Zink space, including the connected components, irreducible components, and intersection behavior of the irreducible components. As an application of the main result, I will also give a description of the supersingular locus of the Shimura variety for the group GU(2,2) over a prime split in the relevant imaginary quadratic field.

- February 5,
**Marty Weissman** (UCSC)

The arithmetic of arithmetic Coxeter groups

**Abstract**: In the 1990s, John H. Conway developed a visual approach to the study of integer-valued binary quadratic forms. His creation, the “topograph,” sheds light on classical reduction theory, the solution of Pell-type equations, and allows tedious algebraic estimates to be simplified with straightforward geometric arguments. The geometry of the topograph arises from a coincidence between the Coxeter group of type (3, ∞) and the arithmetic group PGL2(Z). From this perspective, Conway’s topograph is the first in a series of applications of arithmetic Coxeter groups to number theory. In this talk, I will survey Conway’s results and variations arising from other arithmetic Coxeter groups. Variations are joint work with Christopher D. Shelley and Suzana Milea.

- February 19,
**Holly Swisher** (Oregon State)

Quantum modular forms and singular combinatorial series
**Abstract**: Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of the modular group, up to nontrivial error terms; however, their domains (the complex upper half-plane, and the rationals, respectively) are notably different.

In this talk, we consider the (n+1)-variable combinatorial rank generating function R_n for n-marked Durfee symbols. These are n+1 dimensional multisums for n>1, and specialize to the ordinary two-variable partition rank generating function when n=1. The mock modular properties of R_n for various n and fixed parameters x_i have been previously studied by Bringmann and Ono; Bringmann; Bringmann, Garvan, and Mahlburg; and Folsom and Kimport. The quantum modular properties of R_1 follow from existing results. In our work, we prove that the combinatorial generating function R_n is a quantum modular form when viewed as a function of rationals.

This work is joint with Amanda Folsom, Min-Joo Jang, and Sam Kimport.

- February 26,
**John Bergdall** (Bryn Mawr College)

Upper bounds for constant slope p-adic families of modular forms
**Abstract**: This talk is concerned with the radius of convergence of p-adic families of modular forms — q-series over a p-adic disc whose specialization to certain integer points is the q-expansion of a classical Hecke eigenform of level p. Numerical experiments by Gouvêa and Mazur in the nineties predicted the general existence of such families but also suggested, in spirit, the radius of convergence in terms of an initial member. Buzzard and Calegari showed, ten years later, that the Gouvêa–Mazur prediction was false. It has since remained open question how to salvage it. Here we will present some recent theoretical results towards such a salvage, backed up by numerical data.

- March 5,
**Frank Thorne** (University of South Carolina)
- March 5,
**Michael Harris** (Columbia University)

Note: Speaking in ALGEBRA SEMINAR (4 pm, 210 Deady)

See Algebra Seminar page for details.

### Spring Quarter, 2019

- April 2,
**Charlotte Chan** (Princeton University)
- April 16,
**Catherine Hsu** (University of Bristol)
- May 21,
**William Chen** (McGill University)
- May 28,
**Derek Garton** (Portland State University)

**Previous years**: 2018 2017 2016