Calendar
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
Last modified February 24, 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
Hypergeometric functions over finite fields
In this talk, I will explain about Otsubo’s definition of hypergeometric functions over finite fields, and I will introduce how the confluent hypergeometric functions over finite fields are useful by showing a transformation formula for Appell–Lauricella functions over finite fields. If time allows, I will introduce my recent work on relations between hypergeometric functions and algebraic varieties.
Posted January 28, 2025
Last modified March 12, 2025
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Eun Hye Lee, Texas Christian University
Automorphic form twisted Shintani zeta functions over number fields
In this talk, we will be exploring the analytic properties of automorphic form twisted Shintani zeta functions over number fields. I will start by stating some basic facts from classical Shintani zeta functions, and then we will take a look at the adelic analogues of them. Joint with Ramin Takloo-Bighash.
Posted March 31, 2025
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Be'eri Greenfeld , University of Washington
Complexity and Growth of Infinite Words and Algebraic Structures
Given an infinite word (for example, 01101001$\ldots$), its complexity function counts, for each n, the number of distinct subwords of length n. A longstanding open problem is the "inverse problem": Which functions $f:\mathbb N\to \mathbb N$ arise as complexity functions of infinite words? We resolve this problem asymptotically, showing that, apart from submultiplicativity and a classical obstruction found by Morse and Hedlund in 1938, there are essentially no further restrictions. We then explore parallels and contrasts with the theory of growth of algebras, drawing on noncommutative constructions associated with symbolic dynamical systems.
Posted January 26, 2025
Last modified April 14, 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
How might we generalize the Kronecker-Weber Theorem?
Hilbert's Twelfth Problem asks for a generalization of the Kronecker-Weber Theorem: cyclotomic units generate the abelian extensions of $\mathbb{Q}$, but what about for other ground fields? We consider a number field $K$ with exactly one complex embedding. In the 1970s, Stark conjectured formulas for (the absolute values of) units inside abelian extensions of $K$. We refine Stark's conjectures with a proposed formula for the units themselves, not just their absolute values.
Posted April 14, 2025
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Qing Zhang, University of California Santa Barbara
Realizing Modular Data from Centers of Near-Group Categories
In this talk, I will discuss modular data arising from the Drinfeld centers of near-group categories. The existence of near-group categories of type $G+n$ can be established by solving a set of polynomial equations introduced by Izumi; a different set of equations, also due to Izumi, can then be used to compute the modular data of their Drinfeld centers. Smaller-rank modular categories can often be obtained from these centers via factorization and condensation. After introducing the background of this framework, I will show the existence of a near-group category of type $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z} + 16$ and explain how the modular data of its Drinfeld center can be computed. I will then show that modular data of rank 10 can be obtained via condensation of its Drinfeld center and present an alternative realization of this data through the Drinfeld center of a fusion category of rank 4. Finally, I will discuss the modular data of the Drinfeld center of a near-group category of type $\mathbb{Z}/8\mathbb{Z} + 8$ and demonstrate that the non-pointed factor of its condensation coincides with the modular data of the quantum group category $C(g_2, 4)$. This talk is based on joint work with Zhiqiang Yu.
Posted April 16, 2025
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Andrew Riesen, MIT
Orbifolds of Pointed Vertex Algebras
We will discuss the interplay of tensor categories $C$ with some group action $G$ and orbifolds $V^G$ of vertex operator algebras (VOAs for short). More specifically, we will show how the categorical structure of $\mathrm{TwMod}_G V$ allows one to not only simplify previous results done purely through VOA techniques but vastly extend them. One such example is the Dijkgraaf-Witten conjecture, now a theorem, which describes how the category of modules of a holomorphic orbifold should look like. Additionally, our techniques also allow us to expand the modular fusion categories known to arise from VOAs, we show that every group-theoretical fusion category comes from a VOA orbifold. This talk is based on joint work with Terry Gannon.