Passcode: Numerical Value of 1^3+12^3. (i.e. Ramanujan's taxi-cab number)
Spring 2021 Schedule
- February 5th: Jesse Thorner (U. Illinois, Urbana-Champaign)
An approximate form of Artin's holomorphy conjecture and nonvanishing of Artin L-functions
Abstract: The speaker 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.
- February 12th: Martin Raum (Chalmers U)
Divisibility questions on the partition function and their connection to modular forms
Abstract. The partition function records the number of ways an integer can be written as a sum of positive integers. Already studied by Euler, it has turned out to be a great source of inspiration in the theory of modular forms over the course of the past century. This development was ignited by Ramanujan. At at a time when it was a challenge to merely calculate values of the partition function, he anticipated divisibility properties of astonishing regularity. We will explain some of the ideas that emerged from Ramanujan's conjectures and some of their modern manifestations. Many of these are connected to modular forms and via these to Galois representations. They help us to understand in an increasingly precise sense how frequently Ramanujan's divisibility patterns and their generalizations occur.
- February 19th: Maryam Khaqan (Emory U)
Elliptic curves and moonshine
Abstract: The study of moonshine originated from 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 algebraic structures that these coincidences reflect. We prove the existence of one such structure, a module for Thompson's sporadic simple group, whose graded traces are specific half-integral weight weakly holomorphic modular forms. We then use this module to study the ranks of certain families of elliptic curves. In particular, 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 other families of elliptic curves. Time permitting, we will talk about some of these results as well.
- February 26th: Jennifer Balakrishnan (Boston U)
Rational points on curves and quadratic Chabauty
Abstract: Let C be a smooth projective curve defined over the rational numbers with genus at least 2. It was conjectured by Mordell in 1922 and proved by Faltings in 1983 that C has finitely many rational points. However, Faltings' proof does not give an algorithm for finding these points, and in practice, given a curve, provably finding its set of rational points can be quite difficult. In the case when the Mordell--Weil rank of the Jacobian of C is less than the genus, the Chabauty--Coleman method can be used to find rational points, using the construction of certain p-adic line integrals. Nevertheless, the situation in higher rank is still rather mysterious. I will describe the quadratic Chabauty method (developed in joint work with N. Dogra, S. Müller, J. Tuitman, and J. Vonk), which can apply when the rank is equal to the genus. I will also highlight some examples of interest, from the time of Diophantus to the present day.
- March 5th: Jonathan Love (Stanford)
Explicit Rational Equivalences of Zero-cycles on Surfaces
Abstract: The Chow group of zero-cycles on a smooth projective surface X is obtained by taking the free abelian group generated by closed points on X, and declaring two elements (“zero-cycles”) to be equal if their difference is a sum of divisors of rational functions on curves in X; in this setting we say the zero-cycles are “rationally equivalent.” These Chow groups are notoriously difficult to compute; while a set of conjectures due to Bloch and Beilinson predict certain relations must hold in these groups when X is defined over a number field, there are very few non-trivial cases in which these relations have been proven to hold. In this talk, I will discuss several techniques that can be used to compute rational equivalences exhibiting some of the expected relations, in the case that X is a product of two elliptic curves over Q.
- March 12th: Liyang Yang (Cal Tech)
Average Central L-values on U(2,1)$\times$ U(1,1), Nonvanishing and Subconvexity
Abstract: In this talk, we study an average of automorphic periods on $U(2,1)\times U(1,1).$ We also compute local factors in Ichino-Ikeda formulas for these periods to obtain an explicit asymptotic expression. Combining them together we would deduce some important properties of central $L$-values on $U(2,1)\times U(1,1)$ over a family of holomorphic cusp forms: the first moment, nonvanishing and subconvexity. This is joint work with Philippe Michel and Dinakar Ramakrishnan.
- March 19th: Wei-Lun Tsai (University of Virginia)
Malle's Conjecture for GxA, with G=S3, S4, S5
Abstract: Malle's Conjecture predicts the asymptotic distribution of number fields with fixed Galois group. The conjecture remains wide open. Here we prove several infinite families of the conjecture. This is joint work with R. Masri, F. Thorne, and J. Wang.
- March 26th: Vandita Patel (U Manchester)
- April 2nd: Michael Bennett (U British Columbia)
- April 9th: Jan-Willem van Ittersum (Max Planck, Bonn)
- April 16th: Levent Alpoge (Columbia U)
- April 23rd: Caroline Turnage-Butterbaugh (Carleton)
- April 30th: Humberto Diaz (Washington Univ)
- May 7th: Alex Dunn (Cal Tech)