# Vanderbilt Number Theory Seminar, Fall 2020

This fall 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.  Several of the talks are planned to tie in with the themes of upcoming mock theta conference being held this Spring.

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

Date Speaker, Title, Abstract
Aug 18 Tu Aug 18, 2020 @ 11:00am central time

##### Dennis Stanton (University of Minnesota) – Historical remarks and recent conjectures for integer partitions.

I will concentrate on two areas:

(1) ranks, cranks, and the Ramanujan congruences for p(n),

(2) the Rogers-Ramanujan identities and MacMahon’s combinatorial versions.

Several open questions will be presented.

The slides can be found here.

Aug 25 Tu Aug 25, 2020 @ 11:00am central time
Martin Raum (Chalmers University of Technology and University of Gothenburg) – Relations among Ramanujan-type Congruences.
We present a new framework to access relations among Ramanujan-type congruences of a weakly holomorphic modular form. The framework is strong enough to apply to all Shimura varieties, and covers half-integral weights if unary theta series are available. We demonstrate effectiveness in the case of elliptic modular forms of integral weight, where we obtain a characterization of Ramanujan-type congruences in terms of Hecke congruences. Finally, we showcase concrete computer calculations, exploring the information encoded by our framework in the case of elliptic modular forms of half-integral weight. This leads to an unexpected dichotomy between Ramanujan-type congruences found by Atkin and by Ono, Ahlgren-Ono.The slides can be found here.
Sep 1 Tu Sep 1, 2020 @ 11:00am central time
Maryam Khaqan (Emory University) – Elliptic Curves and Moonshine.

Moonshine began as a series of numerical coincidences connecting finite groups to modular forms. It has since evolved into a rich theory that sheds light on the underlying structures that these coincidences reflect.
We prove the existence of one such structure, a module for the Thompson group, whose graded traces are specific half-integral weight weakly holomorphic modular forms. We then proceed to use this module to study the ranks of certain families of elliptic curves. This serves as an example of moonshine being used to answer questions in number theory.
This talk is based on arXiv: 2008.01607, where we classify all such Thompson-modules where the graded dimension is a specific weakly-holomorphic modular form and prove more subtle results concerning geometric invariants of certain families of elliptic curves. Time permitting, we will talk about some of these results as well.
The slides can be found here.
Sep 8 Tu Sep 8, 2020 @ 11:00am central time
Nicolas Allen Smoot (RISC) – Partition congruences and the localization method.  A notable problem in partition theory is the study of infinite families of partition congruences modulo powers of a prime.  It has recently been discovered that there exist congruence families, associated with a modular curve of genus 0, for which the traditional methods of proof fail.  One such congruence family is related to the spt analogue of the omega mock theta function.  We recently gave a proof of this congruence family by a new method, based on the manipulation of a localized polynomial ring, rather than by studying $$\mathbb{Z}[X]$$ via the more classical methods.  We will give a brief outline of this method, its surprisingly unique characteristics, and its potential for future work.The slides can be found here.
Sep 15 Tu Sep 15, 2020 @ 11:00am central time
Walter Bridges (Louisiana State University) – Statistics for partitions and unimodal sequences.

The study of the asymptotic distribution of statistics for partitions lies at a crossroads of classical methods and the more recent probabilistic framework of Fristedt and others.  We discuss two results—one that uses the probabilistic machinery and one that calls for a more direct “elementary” method.
We first review Fristedt’s conditioning device and, following Romik, implement a similar construction to give an asymptotic formula for distinct parts partitions of $$n$$ with largest part bounded by $$t\sqrt{n}$$.  We discuss the intuitive advantages of this approach over a classical circle method/saddle-point method proof.
We then turn to unimodal sequences, a generalization of partitions where parts are allowed to increase and then decrease.  We use an elementary approach to prove limit shapes for the diagrams of strongly, semi-strict and unrestricted unimodal sequences.  We also recover a limit shape for overpartitions via a simple transfer.
The slides can be found here.
Sep 22 Tu Sep 22, 2020 @ 11:00am central time
Gene Kopp (University of Bristol) – Indefinite zeta functions.
Indefinite theta functions were introduced by Sander Zwegers in his thesis, in which they are used to generalize and explain the remarkable properties of Ramanujan’s mock theta functions. In this talk, we will discuss the Mellin transforms of indefinite theta functions, which we call indefinite zeta functions. Indefinite zeta functions satisfy a functional equation and live in a continuous parameter space. Special points in this parameter space yield arithmetically interesting zeta functions, such as certain differences of ray class zeta functions of real quadratic fields. Generally, however, indefinite zeta functions are not Dirichlet series but have a series expansion involving hypergeometric functions. We prove a Kronecker limit formula in dimension 2 for indefinite zeta functions as s=0, which specializes to a new analytic formula for Stark class invariants.The slides can be found here.
Sep 29 Tu Sep 29, 2020 @ 11:00am central time
Hannah Burson (University of Minnesota) – Mock theta functions, false theta functions, and weighted odd Ferrers diagrams.
Odd Ferrers diagrams are an analogue of integer partitions that were first introduced by George Andrews as a combinatorial interpretation of $$\displaystyle q\omega(q)=\sum_{n=0}^\infty \frac{q^{n+1}}{(q;q^2)_{n+1}}$$. In this talk, we explore the generating functions that result from attaching various weights to these diagrams.  In the process, we give new combinatorial interpretations of some of Ramanujan’s false theta function identities and two second-order mock theta functions.
Oct 6 Tu Oct 6, 2020 @ 11:00am central time
Nikos Diamantis (The University of Nottingham) – Modular iterated integrals associated with cusp forms.
We show how to produce new types of modular iterated integrals originating in cusp forms and are thus more likely to have arithmetic applications. Further, we introduce a new approach to the characterisation of modular iterated integrals as invariant versions of (linear combinations of) iterated integrals formed by modular forms. These constructions are based on an extension of higher-order modular forms which is of independent interest.The slides are here.
Oct 13 Tu Oct 13, 2020 @ 11:00am central time
Jeremy Rouse (Wake Forest University) – Integers represented by positive-definite quadratic forms and Petersson inner products.
We give a survey of results about the problem of determining which integers are represented by a given quaternary quadratic form Q. A necessary condition for Q(x1,x2,x3,x4) to represent n is for the equation Q(x1,x2,x3,x4) = n to have a solution with x1,x2,x3,x4 in Z_p for all p. But even when n is sufficiently large, this is not sufficient for Q to represent n. The form Q is anisotropic at the prime p if for x1,x2,x3,x4 in Z_p, Q(x1,x2,x3,x4) = 0 implies that x1=x2=x3=x4=0. Suppose that A is the Gram matrix for Q and D(Q) = det(A). We show that if n >> D(Q)^{6+\epsilon}, n is locally represented by Q, but Q fails to represent n, then there is an anisotropic prime p so that p^2 | n and np^{2k} is not represented by Q for any k >= 1. We give sharper results when D(Q) is a fundamental discriminant and discuss applications to universality theorems like the 15 and 290 theorems of Bhargava and Hanke.The slides can be found here.
Oct 27 Tu Oct 27, 2020 @ 11:00am central time
Robert Schneider (University of Georgia) – A multiplicative theory of (additive) partitions.
Much like the natural numbers $$\mathbb N$$, the set $$\mathcal P$$ of integer partitions ripples with interesting patterns and relations. Now, Euler’s product formula for the zeta function as well as his generating function formula for the partition function $$p(n)$$ share a common theme, despite their analytic dissimilarity: expand a product of geometric series, collect terms and exploit arithmetic structures in the terms of the resulting series. Given this correspondence, one wonders: might some theorems of classical multiplicative number theory arise as images in $$\mathbb N$$ of greater algebraic or set-theoretic structures in $$\mathcal P$$, and vice versa?We show that many well-known objects from elementary and analytic number theory can be viewed as special cases of phenomena in partition theory such as: a multiplicative arithmetic of partitions that specializes to much of elementary number theory; a class of “partition zeta functions” containing the Riemann zeta function and other Dirichlet series (as well as exotic non-classical cases); partition-theoretic methods to compute arithmetic densities of subsets of $$\mathbb N$$ as limiting cases of $$q$$-series; and other phenomena at the intersection of the additive and multiplicative branches of number theory.The slides can be found here.
Nov 3 Tu Nov 3, 2020 @ 11:00am central time
Madeline Locus Dawsey (The University of Texas at Tyler) – Modular Forms and Ramsey Theory.
For a fixed integer $$k \geq 2$$, let $$q$$ be a prime power such that $$q=1 \pmod{k}$$ if $$q$$ is even, or $$q = 1 \pmod{2k}$$ if $$q$$ is odd. The generalized Paley graph $$G_k(q)$$ is the graph with vertex set $$\mathbb{F}_q$$ and edge set $$\{ab : a-b \text{ is a k-th power residue modulo} \ q\}$$. We give a formula in terms of finite field hypergeometric functions for the number of complete subgraphs of order four contained in $$G_k(q)$$, and a formula in terms of Jacobi sums for the number of complete subgraphs of order three contained in $$G_k(q)$$, for all k. This generalizes work of Evans, Pulham, and Sheehan on the Paley graph with $$k=2$$. We apply these formulas to obtain lower bounds for the multicolor diagonal Ramsey numbers $$R_k(4)$$ and $$R_k(3)$$. We also examine deep connections to Fourier coefficients of modular forms and rational points on elliptic curves, when q is prime.The slides can be found here.
Nov 9 M Nov 9, 2020 @ 10:00am central time
William Craig (The University of Virginia) – Variants of Lehmer’s Conjecture.
In the spirit of Lehmer’s unresolved speculation on the nonvanishing of Ramanujan’s tau-function, it is natural to ask whether a fixed integer is a value of $$\tau(n)$$ or is a Fourier coefficient $$a_f(n)$$ of any given newform $$f(z)$$. We offer a method, which applies to newforms with integer coefficients and trivial residual mod $$2$$ Galois representation, that answers this question for odd integers. We determine infinitely many spaces for which the primes $$3 \leq \ell \leq 37$$ are not absolute values of coefficients of newforms with integer coefficients.We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes $$\ell$$, we prove that $$\pm \ell m$$ is not a coefficient of any such newform $$f$$ with weight $$2k > M_{\pm}(\ell,m) = O_{\ell}(m)$$ and even level coprime to $$\ell$$, where $$M_{\pm}(\ell,m)$$ is effectively computable.The slides can be found here.
Nov 17 Tu Nov 17, 2020 @ 11:00am central time
Robert Osburn (University College Dublin) – Generalized Fishburn numbers and torus knots.

The Fishburn numbers are a sequence of positive integers with numerous combinatorial interpretations and interesting asymptotic properties. In 2016, Andrews and Sellers initiated the study of arithmetic properties of these numbers. In this talk, we discuss a generalization of this sequence using knot theory. This is joint work with Colin Bijaoui (McMaster), Hans Boden (McMaster), Beckham Myers (Harvard),  Will Rushworth (McMaster), Aaron Tronsgard (Toronto) and Shaoyang Zhou (Vanderbilt).
The slides can be found here.
Nov 24 Tu Nov 24, 2020 @ 11:00am central time
Ankush Goswami (RISC) – Arithmeticity and quantum modularity for generalized Kontsevich-Zagier strange series.
There has been significant recent interest in the arithmetic
properties of the coefficients of $$F(1-q)$$ and
$$\mathscr{F}_t(1-q)$$ where $$F(q)$$ is the Kontsevich-Zagier strange
series and $$\mathscr{F}_t(q)$$ is the strange series associated to a family
of torus knots. In this talk, we discuss prime power congruences for two
families of generalized Fishburn numbers and the quantum modularity of
$$\mathscr{F}_t(q)$$.
Dec 1 Tu Dec 1, 2020 @ 11:00am central time
Lola Thompson (Utrecht University) – Counting quaternion algebras, with applications to spectral geometry.
We will introduce some classical techniques from analytic number theory and show how they can be used to count quaternion algebras over number fields subject to various constraints. Because of the correspondence between maximal subfields of quaternion algebras and geodesics on arithmetic hyperbolic manifolds, these counts can be used to produce quantitative results in spectral geometry. This talk is based on joint work with B. Linowitz, D. B. McReynolds, and P. Pollack.
Dec 8 Tu Dec 8, 2020 @ 11:00am central time
Amanda Folsom (Amherst College) – Eisenstein series, cotangent-zeta sums, knots, and quantum modular forms.
Quantum modular forms, defined in the rationals Q, transform like modular forms do on the upper half plane H, up to suitably analytic error functions.  In this talk we give frameworks for two different examples of quantum modular forms originally due to Zagier:  the Dedekind sum, and a certain q-hypergeometric sum due to Kontsevich.  For the first, we extend work of Bettin and Conrey and define twisted Eisenstein series, study their period functions, and establish quantum modularity of certain cotangent-zeta sums.  For the second, we discuss results due to Hikami, Lovejoy, the author, and others, on quantum modular and quantum Jacobi forms ultimately related to colored Jones polynomials for a certain family of knots.