# Vanderbilt University Number Theory Seminar

This spring all seminars will be held on Zoom.  There will also be a 30 minute “coffee break” on Zoom after each talk.  Recordings of the talks will be available here.

The Zoom link for the seminar is here. The password is the weight of the Delta function.

Date Speaker, Title, Abstract
Feb 3 W Feb 3, 2021 @ 11:00am central time

##### Jesse Thorner (University of Illinois) – An approximate form of Artin’s holomorphy conjecture and nonvanishing of Artin L-functions.

I will present some recent work with Robert Lemke Oliver
and Asif Zaman in which we noticeably expand the region in which
almost all Artin L-functions in certain families are holomorphic and
nonvanishing.  This combines Galois theory, character theory, and
analytic number theory.  These results are motivated by a wide range
of classically flavored applications; I will focus on applications to
the study of class groups of high-degree number fields.

Feb 10 W Feb 10, 2021 @ 11:00am central time
Sarah Peluse (Princeton/IAS) – Modular zeros in the character table of the symmetric group.
In 2017, Miller conjectured, based on computational evidence, that for any fixed prime $$p$$ the density of entries in the character table of $$S_n$$ that are divisible by $$p$$ goes to $$1$$ as $$n$$ goes to infinity. I’ll describe a proof of this conjecture, which is joint work with K. Soundararajan. I will also discuss the (still open) problem of determining the asymptotic density of zeros in the character table of $$S_n$$, where it is not even clear from computational data what one should expect.
Feb 17 W Feb 17, 2021 @ 11:00am central time

##### Joshua Males (University of Cologne) – Cycle integrals, theta lifts, and modular forms.

Cycle integrals are intricately linked to many areas of maths. For example, they encode special values of L-functions, give loop amplitudes in string theory, and appear in algebraic geometry. They can often be realised as certain theta lifts. In this talk I’ll give an overview of recent developments in the use of generalised modular forms in determining rationality results of such cycle integrals. Inspired by breakthrough works of Bringmann-Kane-Kohnen and Bruinier-Ehlen-Yang, I will describe some new results on a certain theta lift, and its relationship to cycle integrals, and how it can be realised in terms of coefficients of generalised modular forms. For example, we see how one can recover relationships to Hurwitz class numbers, or the classical spt partition function.

I will also briefly discuss some related ongoing and future topics. Parts of this talk are based on work with Alfes-Neumann, Bringmann, and Schwagenscheidt as well as Scharf and Schwagenscheidt.

Feb 24 W Feb 24, 2021 @ 10:00am central time

##### Kathrin Bringmann (University of Cologne) – False theta functions and their modularity properties.

In my talk I will explain how to embed false theta functions into a modular framework and discuss applications.

Mar 17 W Mar 17, 2021 @ 11:00am central time

##### Jan-Willem Van Ittersum (Utrecht University) – Partitions and quasimodular forms: variations on the Bloch-Okounkov theorem.

Partitions of integers and (quasi)modular forms are related in many ways. We discuss a connection made by a certain normalized generating series of functions f on partitions, called the q-bracket of f. There are many families of functions on partitions, such as (i) the shifted symmetric functions, (ii) their p-adic generalizations, (iii) the weighted t-hook functions and (iv) symmetric functions on partitions, for which the corresponding q-brackets are quasimodular forms. We explain how these four examples can be traced back to the generating series of shifted symmetric functions. The main technical tool for doing so is the study of the Taylor coefficients of strictly meromorphic quasi-Jacobi forms around rational lattice points.

Mar 24 W Mar 24, 2021 @ 11:00am central time
Ling Long (LSU) – A Whipple formula revisited.

A well-known formula of Whipple relates certain hypergeometric values $$_7F_6(1)$$ and $$_4F_3(1)$$. In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data $$HD$$, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple’s formula when the hypergeometric data $$HD$$ are primitive and defined over rationals. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms.  We further relate the  hypergeometric values $$_7F_6(1)$$ in Whipple’s formula to the  periods of modular forms.
This is a joint project with Wen-Ching Winnie Li and Fang-Ting Tu. Most of the talk will be on a general background about how different hypergeometric aspects can be fit together.  It should be accessible to graduate students.
Mar 31 W Mar 31, 2021 @ 11:00am central time
Shashika Petta Mestrige (LSU) – Congruences for some partition functions modulo prime powers.

Ramanujan, Watson, Atkin, Gordon, and Hughes used modular functions and modular equations to prove remarkable congruences of the partition function $$p(n)$$ and multi-partitions $$p_k(n)$$. By extending their ideas, we proved the congruences for two parameter family of partitions $$p_{[1^c\ell^d]}(n)$$ modulo powers of $$\ell$$ where $$\ell$$ is a prime $$(5\leq\ell\leq17)$$. We define these partitions by

$$\displaystyle{\sum_{n=0}^{\infty}p_{[1^c{\ell}^d]}(n)q^n=\prod_{n=1}^{\infty}\frac{1}{(1-q^n)^c(1-q^{\ell n})^d}.}$$

Then we used them to derive congruences and incongruences for $$\ell$$-regular partitions, $$\ell$$-core partitions, and $$\ell$$-colored generalized Frobenius partitions.

Then we investigated the congruences modulo arbitrary prime powers by studying a $$\ell$$-adic module associated to the partitions $$p_{[1^c\ell^d]}(n)$$. Our work has been inspired by the work of Folsom-Kent-Ono-Boylan-Webb.

Apr 7 W Apr 7, 2021 @ 11:00am central time

##### Wei-Lun Tsai (University of Virginia) – Equidistribution of Fourier coefficients of weak Maass forms.

In this talk, I will discuss recent work in which we show that the normalized Fourier coefficients of weak Maass forms of prime level $$p$$ become equidistributed on $$[-1,1]$$ as $$p \rightarrow \infty$$. For integral weight forms, these coefficients are equidistributed with respect to the Sato-Tate measure, while for half-integral weight forms, these coefficients are equidistributed with respect to the arc length measure. The proofs involve a blend of geometric and analytic methods. This is joint work with Riad Masri.

Apr 14 W Apr 14, 2021 @ 11:00am central time
Bernhard Heim (RWTH Aachen University) – Rota’s vision and the Lehmer conjecture.
Lehmer’s conjecture on the non-vanishing of the Ramanujan tau-function is framed in the context of the root distribution of certain families of polynomials. This includes D’Arcais, Nekrasov-Okounkov hook length formula, Chebychev and Laguerre polynomials, and  generalizations of the Bessenrodt-Ono inequality.  We also report on recent progress towards the Chern-Fu-Tang conjecture. This is joint work with Neuhauser.The slides can be found here.
Apr 21 W Apr 21, 2021 @ 11:00am central time

##### Soon-Yi Kang (Kangwon National University) – Arithmetic properties of the Fourier coefficients of weakly holomorphic modular functions of arbitrary level
The canonical basis of the space of modular functions on the modular group of genus zero form a Hecke system.  From this fact, many important properties of modular functions were derived.
In this talk, we show that the Niebur-Poincare basis of the space of Harmonic Maass functions also forms a Hecke system. As its applications, we show several arithmetic properties of modular functions on the higher genus modular curves such as divisibility of Fourier coefficients of modular functions of arbitrary level and arithmetic of divisor polar harmonic Maass forms.
This is a joint work with Daeyeol Jeon and Chang Heon Kim.
Apr 28 W Apr 28, 2021 @ 11:00am central time

##### Jeremy Lovejoy (CNRS Université de Paris) – Parity bias in partitions

By parity bias in partitions, we mean the tendency of partitions to have more odd parts than even parts.   In this talk we will discuss exact and asymptotic results for $$p_{e}(n)$$ and $$p_{o}(n)$$, which denote the number of partitions of n with more even parts than odd parts and the number of partitions of n with more odd parts than even parts, respectively.  We also discuss some open problems, one of which concerns a q-series with an “almost regular” sign pattern, reminiscent of some notorious q-series found in Ramanujan’s lost notebook.  This is joint work with Byungchan Kim and Eunmi Kim.

The slides can be found here.

May 5 W May 5, 2021 @ 11:00am central time
Caroline Turnage-Butterbaugh (Carleton College) – Gaps between zeros of the Riemann zeta-function
Let $$0 < \gamma_1 \le \gamma_2 \le \cdots$$ denote the ordinates of the complex zeros of the Riemann zeta-function function in the upper half-plane. The average distance between $$\gamma_n$$ and $$\gamma_{n+1}$$ is $$2\pi / \log \gamma_n$$ as $$n\to \infty$$. An important goal is to prove unconditionally that these distances between consecutive zeros can much, much smaller than the average for a positive proportion of zeros. We will discuss the motivation behind this endeavor, progress made assuming the Riemann Hypothesis, and recent work with A. Simonič and T. Trudgian to obtain an unconditional result that holds for a positive proportion of zeros.
May 7 F May 7, 2021 @ 11:00am central time
Min Lee (University of Bristol) – Effective equidistribution of rational points on expanding horospheres

The main purpose of this talk is to provide an effective version of a result due to Einsiedler, Mozes, Shah and Shapira, on the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classification theorems due to Ratner. Instead, we pursue an alternative strategy based on spectral theory of automorphic forms, Fourier analysis and Weil’s bound for Kloosterman sums which yields an effective estimate on the rate of convergence in the space of (d+1)-dimensional Euclidean lattices.
This is a joint work with D. El-Baz, B. Huang and J. Marklof.
May 12 W May 12, 2021 @ 11:00am central time

##### Matthew Just (University of Georgia) – Partition Eisenstein series and semi-modular forms

We identify a class of “semi-modular” forms invariant on special subgroups of $$GL_{2}(\mathbb{Z})$$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein- like series summed over integer partitions, and use it to construct families of semi- modular forms. We ask whether other examples exist, and what properties they all share. This is joint work with Robert Schneider.