# Number Theory Seminar 2018-2019

This seminar is held on Tuesdays at 1pm in 119 Fenton Hall unless otherwise stated.

### Spring Quarter, 2019

- April 16,
**Catherine Hsu** (University of Bristol)

The Eisenstein ideal and R=T Theorems
**Abstract**: In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk, we re-examine Eisenstein congruences, incorporating a notion of “depth of congruence,” in order to understand the local structure of Eisenstein ideals associated to weight 2 cusp forms of squarefree level. We then discuss implications of a non-principal Eisenstein ideal within the context of various R=T theorems.

- April 23,
**Elena Mantovan** (Cal Tech)

Shimura varieties and the Torelli locus
**Abstract**: The Schottky problem is a classical and fundamental question in arithmetic algebraic geometry, about the characterization of Jacobian varieties among abelian varieties. This question is equivalent to studying the Torelli locus (i.e., the image of moduli of curves under the Torelli map) inside Siegel modular varieties. In positive characteristics, a first approximation to this problem is understanding the discrete invariants (e.g., p-rank, Newton polygon, Ekedahl–Oort type) occurring for Jacobians of smooth curves. The Coleman–Oort conjecture predicts that if the genus is large, then up to isomorphism, there are only finitely many smooth projective curves over the field of complex numbers, of genus g and Jacobian an abelian variety with complex multiplication. An effective version of the Colemann–Oort conjecture proposes 8 as an explicit lower bound.

After introducing the framework for these problems, I will discuss recent progress towards the Schottky problem in positive characteristics which is inspired by the Coleman–Oort conjecture, and which relies on the study of special subvarieties (a.k.a, Shimura subvarieties) of Siegel varieties.

- May 21,
**William Chen** (McGill University)

Noncongruence modular forms and nonabelian level structures for elliptic curves
**Abstract**: For noncongruence subgroups of SL(2,Z), the classical definition of modular forms still makes sense, and they still admit Fourier expansions at cusps. Their main difference compared to congruence forms is the (non)existence of a good Hecke theory, which is responsible for most of the key arithmetic properties enjoyed by congruence modular forms. In this talk I will explain how even in the absence of a Hecke theory, the Fourier coefficients of noncongruence modular forms have interesting arithmetic properties tied to the arithmetic of nonabelian level structures on elliptic curves.

- May 28,
**Derek Garton** (Portland State University)

Statistics of a-numbers of hyperelliptic curves over finite fields
**Abstract**: We apply a Minkowski-type theorem for function fields to compute the proportion of certain families of hyperellptic curves over finite fields of characteristic three that have any particular a-number. In particular, we compute the number of hyperellptic curves of any genus with maximal a-number over any such field. This work is joint with

Jeffrey Lin Thunder and Colin Weir.

- June 4,
**Joe Webster** (UO)

log-Coulomb gas in a nonarchimedean local field
**Abstract**: Random matrix theorists of the mid-20th century noticed that the eigenvalues of certain Gaussian matrix ensembles act like a “one-dimensional log-Coulomb gas”. This physical model has since become well-understood in the language of Boltzmann statistical mechanics. In this talk we introduce a nonarchimedean analogue, use a combinatorial method to establish basic results, and briefly discuss their physical interpretation. Time permitting, we’ll also discuss how these results simplify in the special case that the gas particles are identical.

### 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.

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

Levels of distribution for prehomogeneous vector spaces
**Abstract**: I will discuss some work with Takashi Taniguchi in proving “levels of distribution for prehomogeneous vector spaces” — basically, cumulative bounds on error terms in arithmetic statistics.

First, I will give an overview of why one might care — what prehomogeneous vector spaces are, what they have to do with arithmetic

statistics, and where these error terms show up. Then, I will outline our method — which involves a little bit of combinatorics, a little

bit of Fourier analysis, and a little bit of algebraic geometry.

- March 5,
**Michael Harris** (Columbia University)

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

See Algebra Seminar page for details.
- March 12,
**Angus McAndrew** (Boston University)

Differential operators on automorphic forms and Galois representations
**Abstract**: When one first learns the theory of modular forms, one is taught that they are certain special complex differentiable functions on the upper half plane. A natural instinct, therefore, is to ask what happens when one goes ahead and differentiates them? This leads to the rich theory of differential operators on Shimura varieties and automorphic forms. We describe some work with Alexandru Ghitza, Ellen Eischen and Elena Mantovan on the analytic continuation of differential operators from the ordinary locus, as well as the relationship with Galois representations.

**Previous years**: 2018 2017 2016