Posted August 19, 2022

Last modified August 26, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Lockett 232 and Zoom
Daniel Fretwell, University of South Wales

Definite orthogonal modular forms: Computations, Excursions and Discoveries

The theory of (positive definite) integral quadratic forms and lattices has a long and rich history. For many years it has been known how to study these objects via their theta series, modular forms whose Fourier coefficients encode arithmetic data. A less well known fact is that isometry classes of lattices (in a genus) can themselves be viewed as automorphic forms, for the corresponding (definite) orthogonal group. These forms also contain a wealth of arithmetic information. In general, algorithms for computing spaces of automorphic forms for higher rank groups are few and far between. However, the case of definite orthogonal groups is concrete enough to be amenable to computation, and provides a significant testing ground for general conjectures in the Langlands Program (e.g. explicit Functoriality). Recently, E. Assaf and J. Voight have developed a new magma package for computing spaces of orthogonal modular forms. We will take a stroll through a zoo of explicit examples computed using this package, outlining links with conjectures of Arthur on endoscopy and discoveries of new Eisenstein congruences. (Joint work with E. Assaf, C. Ingalls, A. Logan, S. Secord and J. Voight)

Posted August 19, 2022

Last modified September 6, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Lockett 232 and Zoom
Michael Allen, Louisiana State University

On some supercongruence conjectures of Long

In 2003, Rodriguez Villegas conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of weight 4 modular forms. Uniform proofs of these supercongruences were given in 2019 by Long, Tu, Yui, and Zudilin. In 2020 Long conjectured a number of further supercongruences for hypergeometric functions of a similar shape. In this talk, we extend the approach of Long, Tu, Yui, and Zudilin towards establishing six of Long's conjectures, and also discuss possible future directions and further generalizations.

Posted August 19, 2022

Last modified September 6, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Lockett 232 and Zoom
Olivia Beckwith, Tulane University

Ramanujan-type congruences for Hurwitz class numbers

The Ramanujan congruences, discovered over a century ago, state that the partition function is annihilated modulo p on a certain arithmetic progression if p is 5, 7, or 11. The work of Ono, Ahlgren, and Treneer shows the coefficients of any weakly holomorphic modular form have infinitely many similar congruence properties. We examine congruences for Hurwitz class numbers, in which case the generating series are mock modular instead of modular. We prove that congruences for Hurwitz class numbers exist on square classes, and we classify the arithmetic progressions appearing in such congruences. This is joint work with Martin Raum and Olav Richter.

Posted August 19, 2022

Last modified November 29, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Lockett 232 and Zoom
Ayla Gafni, University of Mississippi

Uniform distribution and geometric incidence theory

The Szemerédi–Trotter Incidence Theorem, a central result in geometric combinatorics, bounds the number of incidences between $n$ points and $m$ lines in the Euclidean plane. Replacing lines with circles leads to the unit distance problem, which asks how many pairs of points in a planar set of $n$ points can be at a unit distance. The unit distance problem breaks down in dimensions $4$ and higher due to degenerate configurations that attain the trivial bound. However, nontrivial results are possible under certain structural assumptions about the point set. In this talk, we will give an overview of the history of these and other incidence results. Then we will introduce a quantitative notion of uniform distribution and use that property to obtain nontrivial bounds on unit distances and point-hyperplane incidences in higher-dimensional Euclidean space. This is based on joint work with Alex Iosevich and Emmett Wyman.

Posted August 19, 2022

Last modified September 26, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Zoom
Hasan Saad, University of Virginia

Explicit Sato-Tate type distribution for a family of $K3$ surfaces

In the 1960s, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual Sato-Tate for non-CM elliptic curves). In analogy with Birch's result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces $A_{\lambda}(p)$ of a certain family of $K3$ surfaces $X_\lambda$ with generic Picard rank $19$ is the $O(3)$ distribution. This distribution, which we denote by $\frac{1}{4\pi}f(t),$ is quite different from the semicircular distribution. It is supported on $[-3,3]$ and has vertical asymptotes at $t=\pm1.$ Here we make this result explicit. We prove that if $p\geq 5$ is prime and $-3 \leq a \lt b \leq 3$, then $$ \left|\frac{\#\{\lambda\in\mathbb{F}_p :A_{\lambda}(p)\in[a,b]\}}{p}-\frac{1}{4\pi}\int_a^b f(t)dt\right|\leq \frac{110.84}{p^{1/4}}. $$ As a consequence, we are able to determine when a finite field $\mathbb{F}_p$ is large enough for the discrete histograms to reach any given height near $t=\pm1.$ To obtain these results, we make use of the theory of Rankin-Cohen brackets in the theory of harmonic Maass forms.

Posted August 21, 2022

Last modified September 28, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Lockett 232 and Zoom
Paul Pollack, University of Georgia

Some problems on the value-distribution of arithmetic functions

We discuss two strands of questions about the value-distribution of arithmetic functions. In the first half, we consider distribution in arithmetic progressions. For instance, let $A(n)$ denote the sum of the primes dividing $n$ (with multiplicity). I will sketch a proof that the values of $A(n)$, sampled for $n \leq x$ (with $x \to \infty$), are equidistributed $\pmod{q}$ both for every fixed modulus $q$ (as was known already) and for $q$ that grow slowly with $x$. A result about distribution $\pmod{q}$ is really a result about `trailing digits' working in base $q$. The second half of the talk concerns leading digits. After recalling `Benford's Law' I will describe why the leading digits of the divisor function $d(n)$ tend to follow Benford's law but why the leading digits of the sum-of-divisors function $\sigma(n)$ do not. This is joint work with Fai Chandee and Xiannan Li (Kansas State) and Akash Singha Roy (UGA).

Posted August 19, 2022

Last modified September 30, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Lockett 232 and Zoom
Catherine Hsu, Swarthmore College

Explicit non-Gorenstein $R=T$ via rank bounds

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'll explore generalizations of Mazur's result to squarefree level, focusing on recent work, joint with P. Wake and C. Wang-Erickson, about a non-optimal level $N$ that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein. First, we will outline a Galois-theoretic criterion for the deformation ring to be as small as possible, and when this criterion is satisfied, deduce an $R=T$ theorem. Then we'll discuss some of the techniques required to computationally verify the criterion.

Posted August 19, 2022

Last modified October 15, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Lockett 232 and Zoom
Kim Klinger-Logan, Kansas State University

An application of automorphic forms to string theory

Recently, physicists Green, Russo, and Vanhove have discovered solutions to differential equations involving automorphic forms appear as the coefficients to the 4-graviton scattering amplitude in type IIB string theory. We will discuss a particular form of equation that appears in this context and different approaches to the solution. Time permitting, we will also discuss a connection to a shifted convolution sum that appears in this context. This is joint work with Stephen D. Miller, Danylo Radchenko and Ksenia Fedosova.

Posted August 19, 2022

Last modified October 14, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Lockett 232 and Zoom
Lawrence Washington, University of Maryland

Heuristics for anticyclotomic $\mathbb{Z}_p$-extensions

For an imaginary quadratic field, there are two natural $\mathbb{Z}_p$-extensions, the cyclotomic and the anticyclotomic. We'll start with a brief description of Iwasawa theory for the cyclotomic extensions, and then describe some computations for anticyclotomic $\mathbb{Z}_p$-extensions, especially the fields and their class numbers.

Posted October 15, 2022

Last modified October 31, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Lockett 232 and Zoom
Emma Lien, Louisiana State University

Galois Representations and Weight One Eta-Quotients

A classical problem in number theory is to determine the primes $p$ for which a polynomial splits into linear factors modulo $p$. One is then naturally led to consider the Artin representations associated to the polynomial, i.e, the complex representations of the finite Galois group of its splitting field. Serre and Deligne showed that the representations associated to a weight one Hecke eigenforms are Artin representations. Thus, we wish to examine some easily computable examples of weight 1 Hecke eigenforms coming from eta-quotients with the goal of determining the explicit polynomials associated to them. For example, let $f(\tau)=\eta(6\tau)\eta(18\tau)$; then if $a_n$ denotes the $n$-th coefficient in the Fourier expansion of $f$ and $p>3$ is a prime, then $a_p = 2$ if and only if $x^3-2$ splits modulo $p$. In particular, the representations give us information about certain abelian extensions of imaginary quadratic extensions of $\mathbb{Q}$ and we can even express certain cases as a theta series associated to a quadratic form twisted by a grossencharacter.

Posted October 15, 2022

Last modified November 9, 2022

Algebra and Number Theory Seminar Questions or comments?

1:40 pm - 2:30 pm Lockett 241 and Zoom
Abbey Bourdon, Wake Forest University

Sporadic Torsion on Elliptic Curves

An elliptic curve is a curve in projective space whose points can be given the structure of an abelian group. In this talk, we will focus on torsion points, which are points having finite order under this group law. While we can generally determine the torsion points of a fixed elliptic curve defined over a number field, there are several open problems which require controlling the existence of torsion points within infinite families of elliptic curves. Success stories include Merel's Uniform Boundedness Theorem, which states that the order of a torsion point can be bounded by the degree of its field of definition. On the other hand, a proof of Serre's Uniformity Conjecture---which has been open for 50 years---would in particular imply that for sufficiently large primes $p$, there do not exist points of order $p^2$ arising on elliptic curves defined over field extensions of ``unusually low degree." In this talk, I will give a brief introduction to the arithmetic of elliptic curves before addressing the problem of identifying elliptic curves producing a point of large order in usually low degree, i.e., those possessing a sporadic torsion point. More precisely, let $E$ be an elliptic curve defined over a field extension $F/\mathbb{Q}$ of degree $d$, and let $P$ be a point of order $N$ with coordinates in $F$. Such a point is called ``rational" since it is defined over the same field as $E$. We say $P$ is sporadic if, as one ranges over all fields $F/\mathbb{Q}$ of degree at most $d$ and all elliptic curves $E/F$, there are only finitely many elliptic curves which possess a rational point of order $N$. Sporadic pairs $(E,P)$ correspond to exceptional points on modular curves, which are points whose existence is not explained by standard geometric constructions.

Posted October 15, 2022

Last modified November 9, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Lockett 232 and Zoom
Nicole Looper, University of Illinois at Chicago

Diophantine Techniques in Arithmetic Dynamics

This talk will explore some of the most important relationships between Diophantine geometry and arithmetic dynamics. Many questions in arithmetic dynamics are inspired by classical problems in arithmetic geometry, and many dynamical consequences follow from well-known Diophantine inputs such as the abc conjecture. Moreover, ideas drawn from dynamics are often useful in tackling number-theoretic questions. I will give an overview of these links, and then will discuss some concrete illustrative examples. I will also point out some areas of difficulty that appear key to future progress.

Posted November 9, 2022

Algebra and Number Theory Seminar Questions or comments?

4:15 pm - 5:45 pm Lockett 232 and ZoomBARD 1 Lightning Talks

Short talks (up to 10 minutes) by Prerna Agarwal (LSU), Andrea Bourque (LSU), Pranabesh Das (Xavier University of Louisiana), Brian Grove (LSU), Emma Lien (LSU), Evangelos Nastas (State University of New York), Matthias Storzer (Max Planck Institute), and Kalani Thalagoda (University of North Carolina Greensboro)

Posted October 15, 2022

Last modified November 22, 2022

Algebra and Number Theory Seminar Questions or comments?

3:10 pm - 4:00 pm Lockett 232 and Zoom
Louis Gaudet, Rutgers University

The least Euler prime via sieve

Euler primes are primes of the form $p = x^2+Dy^2$ with $D>0$. In analogy with Linnik’s theorem, we can ask if it is possible to show that $p(D)$, the least prime of this form, satisfies $p(D) \ll D^A$ for some constant $A>0$. Indeed Weiss showed this in 1983, but it wasn’t until 2016 that an explicit value for $A$ was determined by Thorner and Zaman, who showed one can take $A=694$. Their work follows the same outline as the traditional approach to proving Linnik’s theorem, relying on log-free zero-density estimates for Hecke L-functions and a quantitative Deuring-Heilbronn phenomenon. In an ongoing work (as part of my PhD thesis) we propose an alternative approach to the problem via sieve methods that (as far as results about zeros of the Hecke $L$-functions) only requires the classical zero-free region. We hope that such an approach may result in a better value for the exponent $A$.