Posted November 13, 2023
Last modified January 21, 2024
Algebra and Number Theory Seminar Questions or comments?
3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Zhongkai Mi, Louisiana State University
The Lowest Discriminant Ideals of Cayley-Hamilton Hopf Algebras
Discriminant ideals for an algebra $A$ module finite over a central subring $C$ are indexed by positive integers. We study the lowest of them with nonempty zero set in Cayley Hamilton Hopf algebras whose identity fibers are basic algebras. Key results are obtained by considering actions of characters in the identity fiber on irreducible modules over maximal ideals of $C$ and actions of winding automorphisms. We apply these results to examples in group algebras of central extensions of abelian groups, big quantum Borel subalgebras at roots of unity and quantum coordinate rings at roots of unity. This is joint work with Quanshui Wu and Milen Yakimov.
Posted November 13, 2023
Last modified February 2, 2024
Algebra and Number Theory Seminar Questions or comments?
3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Rajat Gupta, University of Texas at Tyler
On summation formulas attached to Hecke's functional equation and $p$-Herglotz functions
In this talk, we will review the work of Chandrasekharan and Narasimhan on the theory of Hecke’s functional equation (with one gamma factor) and the summation formulas of various kinds, such as the Voronoi summation formula, the Poisson summation formula, and the Abel-Plana summation formula. We will then give recent developments in this theory followed by some new results and summation formulas in the setting of Hecke’s functional equation analogous to the ones mentioned above. In particular, I will discuss these summation formulas in the case of cusp corms of weight $2k$ attached to the modular group ${\rm SL}_2(\mathbb{Z})$. Finally, I will also talk about on my recent work with Rahul Kumar on Herglotz functions and their analogues.
Posted February 5, 2024
Last modified February 14, 2024
Algebra and Number Theory Seminar Questions or comments?
3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Hasan Saad, University of Virginia
Distributions of points on hypergeometric varieties
In the 1960's, 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., $SU(2),$ the usual Sato-Tate for non-CM elliptic curves). In this talk, we show how the theory of harmonic Maass forms and modular forms allow us to determine the limiting distribution of normalized traces of Frobenius over families of varieties. For Legendre elliptic curves, the limiting distribution is $SU(2),$ whereas for a certain family of $K3$ surfaces, the limiting distribution is $O(3).$ Since the $O(3)$ distribution has vertical asymptotes, we show how to obtain an explicit result by bounding the error. Additionally, we show how to count "matrix" points on these varieties and therefore determine the limiting distributions for these "matrix points."
Posted November 14, 2023
Last modified February 27, 2024
Algebra and Number Theory Seminar Questions or comments?
3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Eleanor McSpirit, University of Virginia
Infinite Families of Quantum Modular 3-Manifold Invariants
In 1999, Lawrence and Zagier established a connection between modular forms and the Witten-Reshetikhin-Turaev invariants of 3-manifolds by constructing q-series whose radial limits at roots of unity recover these invariants for particular manifolds. These q-series gave rise to some of the first examples of quantum modular forms. Using a 3-manifold invariant recently developed Akhmechet, Johnson, and Krushkal, one can obtain infinite families of quantum modular invariants which realize the series of Lawrence and Zagier as a special case. This talk is based on joint work with Louisa Liles.
Posted November 13, 2023
Last modified March 3, 2024
Algebra and Number Theory Seminar Questions or comments?
3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Edmund Yik-Man Chiang, The Hong Kong University of Science and Technology
Discrete special functions: a D-modulus approach to special functions
We show that there is a holonomic D-modules (PDEs) approach to classical special functions, as such both the classical special functions and their difference analogues, some have only been found recently, can be efficiently computed by Weyl-algebraic framework. According to Truesdell, rudiments of algebraic approaches to special functions were already observed by some nineteenth century mathematicians. This algebraic method does not use well-known Lie algebra theory explicity apart from basic knowledge of solving linear PDEs. We illustrate our method with Bessel functions in this talk. We shall also explain the connection with this D-modules approach with recent advances in Nevanlinna theories for difference operators, which have its roots from discrete Painleve equations.
Posted November 14, 2023
Last modified March 26, 2024
Algebra and Number Theory Seminar Questions or comments?
3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Micah Milinovich, University of Mississippi
Biases in the gaps between zeros of Dirichlet L-functions
We describe a family of Dirichlet L-functions that provably have unusual value distribution and experimentally have a significant and previously undetected bias in the distribution of gaps between their zeros. This has an arithmetic explanation that corresponds to the nonvanishing of a certain Gauss-type sum. We give a complete classification of the characters for when these sums are nonzero and count the number of corresponding characters. It turns out that this Gauss-type sum vanishes for 100% of primitive Dirichlet characters, so L-functions in our newly discovered family are rare (zero density set amongst primitive characters). If time allows, I will also describe some newly discovered experimental results concerning a "Chebyshev-type" bias in the gaps between the zeros of the Riemann zeta-function. This is joint work with Jonathan Bober (Bristol) and Zhenchao Ge (Waterloo).
Posted August 21, 2024
Last modified August 28, 2024
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Rahul Kumar, Pennsylvania State University
Period function from Ramanujan's Lost Notebook and Kronecker limit formulas
The Lost Notebook of Ramanujan contains a number of beautiful formulas, one of which can be found on page 220. It involves an interesting function, which we denote as $\mathcal{F}_1(x)$. In this talk, we show that $\mathcal{F}_1(x)$ belongs to the category of period functions as it satisfies the period relations of Maass forms in the sense of Lewis and Zagier. Hence, we refer to $\mathcal{F}_1(x)$ as the Ramanujan period function. The Kronecker limit formulas are concerned with the constant term in the Laurent series expansion of certain Dirichlet series at $s=1$. We will also discuss that $\mathcal{F}_1(x)$ naturally appears in a Kronecker limit-type formula of a certain zeta function.
Posted August 14, 2024
Last modified September 5, 2024
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Heidi Goodson, Brooklyn College, CUNY
An Exploration of Sato-Tate Groups of Curves
The focus of this talk is on families of curves and their associated Sato-Tate groups -- compact groups predicted to determine the limiting distributions of coefficients of the normalized L-polynomials of the curves. Complete classifications of Sato-Tate groups for abelian varieties in low dimension have been given in recent years, but there are obstacles to providing classifications in higher dimension. In this talk I will give an overview of the techniques we can use for some nice families of curves and discuss the ways in which these techniques fall apart when there are degeneracies in the algebraic structure of the associated Jacobian varieties. I will include examples throughout the talk in order to make the results more concrete to those new to this area of research.
Posted August 21, 2024
Last modified October 7, 2024
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Wanlin Li, Washington University in St. Louis
Non-vanishing of Ceresa and Gross–Kudla–Schoen cycles
The Ceresa cycle and the Gross–Kudla–Schoen modified diagonal cycle are algebraic $1$-cycles associated to a smooth algebraic curve with a chosen base point. They are algebraically trivial for a hyperelliptic curve and non-trivial for a very general complex curve of genus $\ge 3$. Given a pointed algebraic curve, there is no general method to determine whether the Ceresa and GKS cycles associated to it are rationally or algebraically trivial. In this talk, I will discuss some methods and tools to study this problem.
Posted August 14, 2024
Last modified October 17, 2024
Algebra and Number Theory Seminar Questions or comments?
2:30 pm Lockett 233 or click here to attend on Zoom
Brian Grove, LSU
The Explicit Hypergeometric Modularity Method
The existence of hypergeometric motives predicts that hypergeometric Galois representations are modular. More precisely, explicit identities between special values of hypergeometric character sums and coefficients of certain modular forms on appropriate arithmetic progressions of primes are expected. A few such identities have been established in the literature using various ad-hoc techniques. I will discuss a general method to prove these hypergeometric modularity results in dimensions two and three. This is joint work with Michael Allen, Ling Long, and Fang-Ting Tu.
Posted August 21, 2024
Last modified October 25, 2024
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Brett Tangedal, University of North Carolina, Greensboro
Real Quadratic Fields and Partial Zeta-Functions
We focus on real quadratic number fields and explain an approach to the partial zeta-functions associated with the various ideal class groups of such fields dating back to the original work of Zagier, Stark, Shintani, David Hayes, and others. Along the way, we will give a brief introduction to Stark's famous first order zero conjecture.
Posted October 8, 2024
Last modified October 30, 2024
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Virtual talk: click here to attend on Zoom
Linli Shi, University of Connecticut
On higher regulators of Picard modular surfaces
The Birch and Swinnerton-Dyer conjecture relates the leading coefficient of the L-function of an elliptic curve at its central critical point to global arithmetic invariants of the elliptic curve. Beilinson’s conjectures generalize the BSD conjecture to formulas for values of motivic L-functions at non-critical points. In this talk, I will relate motivic cohomology classes, with non-trivial coefficients, of Picard modular surfaces to a non-critical value of the motivic L-function of certain automorphic representations of the group GU(2,1).
Posted October 8, 2024
Last modified November 4, 2024
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Michael Allen, Louisiana State University
An infinite family of hypergeometric supercongruences
In a recent series of papers with Brian Grove, Ling Long, and Fang-Ting Tu, we explore the relationship between modular forms and hypergeometric functions in the particular settings of complex, finite, and $p$-adic fields, and unify these perspectives through Galois representations. In this talk, we focus primarily on the $p$-adic aspects, where this relationship arises in the form of congruences between truncated hypergeometric sums and Fourier coefficients of modular forms. Such congruences are predicted to hold modulo $p$ by formal commutative group law, we refer to a congruence modulo a higher power of $p$ as a supercongruence. In this talk, we briefly survey results and methods in the area of supercongruences before establishing an infinite family of supercongruences which hold modulo $p^2$ for all primes in certain arithmetic progressions depending on the parameters of the corresponding hypergeometric functions.
Posted October 8, 2024
Last modified November 18, 2024
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
David Lowry-Duda, ICERM
Murmuration phenomena in number theory
Approximately 2 years ago, a group of number theorists experimenting with machine learning observed unexpected biases in data from elliptic curves. When plotted, these biases loosely resemble gatherings of starlings, leading to the name "murmurations." This now seems to be a very general phenomenon in number theory. Many different families of arithmetic objects exhibit consistent biases. But proving these behaviors has been challenging. In this talk, we'll give several examples of murmuration phenomena, connect these biases to distributions of zeros of L-functions, and describe recent success proving murmurations (especially for modular forms).
Posted August 29, 2024
Last modified December 2, 2024
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Jiuya Wang, University of Georgia
Counterexamples for Turkelli's Modification of Malle's Conjecture
Malle's conjecture gives a conjectural distribution of number fields with bounded discriminant. Klueners gives counterexamples of Malle's conjecture, due to the presence of roots of unity in intermediate fields. These types of counterexamples exists in both global function fields and number fields. Turkelli proposes a modification of Malle's conjecture inspired by a function field analogue. We give counterexamples for Turkelli's modified conjecture. We will also talk about the difference of Malle's conjecture on function fields and number fields.
Posted January 11, 2025
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:20 pm Virtual talk: click here to attend on Zoom
Walter Bridges, University of North Texas.
The proportion of coprime fractions in number fields
The ring $\mathbb{Z}[\sqrt{-5}]$ is often one of the first examples students encounter of a ring that is not unique factorization domain. Relatedly, in the number field $\mathbb{Q}(\sqrt{-5})$, we have $$ \frac{1+\sqrt{-5}}{2}=\frac{3}{1-\sqrt{-5}}. $$ Both fractions are reduced, meaning that numerator and denominator do not share any (non-unit) factors in $\mathbb{Z}[\sqrt{-5}]$. However, neither fraction is coprime, in the sense that the numerator and denominator pair do not generate $\mathbb{Z}[\sqrt{-5}]$. In this talk, we will answer the question of how often this phenomenon occurs. That is, we compute the density, suitably defined, of the set of coprime fractions in the set of all reduced fractions in a generic number field. Our answer for $\mathbb{Q}(\sqrt{-5})$ is 80%. We will begin with a review of algebraic number theory, then discuss our notion of density in number fields. Finally, we will show that the density in question may be computed using well-known properties of Hecke L-functions. We intend this talk to be accessible to beginning graduate students.
Posted January 19, 2025
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Virtual talk: click here to attend on Zoom
Asimina Hamakiotes, University of Connecticut
Abelian extensions arising from elliptic curves with complex multiplication
Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f \geq 1$. Let $E$ be an elliptic curve with complex multiplication by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$, where $j(E)$ is the $j$-invariant of $E$. Let $N\geq 2$ be an integer. The extension $\mathbb{Q}(j(E), E[N])/\mathbb{Q}(j(E))$ is usually not abelian; it is only abelian for $N=2,3$, and $4$. Let $p$ be a prime and let $n\geq 1$ be an integer. In this talk, we will classify the maximal abelian extension contained in $\mathbb{Q}(E[p^n])/\mathbb{Q}$.
Posted January 26, 2025
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Akio Nakagawa, Kanazawa University
TBA
Posted January 26, 2025
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Kairi Black, Duke University
TBA