Philadelphia Area Number Theory Seminar

Shira Gilat, UPenn

Abstract:The algebra of supernatural matrices is a key example in the theory of locally finite central simple algebras.  Supernatural matrices are a minimal solution to the equation of unital algebras M_n(X) ∼= X, which we compare to several similar conditions involving cancellation of matrices. This algebra has appeared under various names before, and it generalizes both McCrimmon's deep matrices algebra and m-petal Leavitt path algebra.

Katy Woo, Princeton University

Abstract: We resolve Manin's conjecture for all Châtelet surfaces over Q (surfaces given by equations of the form x^2 + ay^2 = f(z)) -- we establish asymptotics for the number of rational points of increasing height. The key analytic ingredient is estimating sums of Fourier coefficients of modular forms along polynomial values.

Ian Whitehead, Swarthmore College

Abstract: I will discuss joint work in progress with Will Sawin on function field multiple Dirichlet series constructed based on a set of axioms from algebraic geometry. These series have applications to moments of Dirichlet L-functions for characters of fixed order. We prove functional equations that imply meromorphic continuation for many of these series. The most striking feature of our work is a set of detailed predictions for the symmetries involved in any given moment computation. We find a complete tabulation of moments that are possible to compute via known multiple Dirichlet series methods, including several new moments.

Abstract: Hel Braun was a mathematician whose substantial work continues to impact research today.  Braun's research contributions lie in three areas: classical number theory problems about integers, modular and automorphic forms, and Jordan algebras.  I will introduce how each of these seemingly distinct topics led Braun naturally to the next, and I will highlight the ongoing impact of some of Braun's most significant contributions.  Reasons for interest today are largely different from those that motivated Braun and were likely unforeseen at the time.  I will also try to address apparent anomalies, like why you probably are unfamiliar with Braun despite the lasting impact of this work.  This colloquium-level talk is intended for a broad mathematical audience.  If your response to the title was "Who's Hel Braun?" this talk is for you.  If you already know who that is but asked "But why Hel Braun?" it is also for you.

Abstract: In this talk, we will discuss some recent applications of machine learning to number theory. In particular, we will discuss murmurations of Arithmetic L-functions and experiments on the Tate–Shafarevich groups. We will also introduce the recent results of applying transformer models to predict information about Frobenius traces a_p from elliptic curves given information about other traces a_q. Our experiments reveal that these models achieve high accuracy, even in the absence of explicit number-theoretic tools like functional equations of L-functions. We also present partial interpretability findings on the patterns learned by the machine learning models. No prior knowledge of machine learning is assumed.

Abstract: Given a Hilbert modular form $f$ of weight two over a totally real field $F$, we can associate to it a finite module $\Phi(f)$ known as the congruence module for $f$, which measures the congruences that $f$ satisfies with other forms. When $f$ is transferred to a quaternionic modular form $f_D$ over a quaternion algebra $D$ via the Jacquet-Langlands correspondence, we can similarly define a congruence module $\Phi(f_D)$ for $f_D$. Pollack and Weston proposed a quantitative relationship between the sizes of $\Phi(f)$ and $\Phi(f_D)$, expressed in terms of invariants associated to $f$ and $D$.

In this talk, I will outline the ideas underlying the proof of this relationship. The approach combines a method of Ribet and Takahashi with new techniques introduced by Böckle, Khare, and Manning.

Abstract: Selmer groups are a cohomological tool for finding rational points on elliptic curves and abelian varieties in general. I'll explain how they work in down to earth terms and give some applications to concrete questions such as: What is the average rank of an elliptic curve over Q? How many integers can be written as a sum of two rational cubes? Is there an algorithm to determine solubility of diophantine equations over a given finitely generated ring? This is also setup for the afternoon talk, where we will apply the Selmer framework to study class groups of number fields.

Abstract: When you want to solve a Diophantine equation in practice, one of the most powerful modern techniques is the Chabauty—Kim method, which proceeds by constructing p-adic functions vanishing on the rational solutions of your equation. This talk will focus on giving specific examples where we can understand these functions quite explicitly. First, we will explain how to produce abelian integrals vanishing on rational points: the so-called Chabauty—Coleman method. Then we will explain one example going beyond the abelian level: the dilogarithm for once-punctured elliptic curves. This latter is the historical starting point of the quadratic Chabauty method.

Abstract: Heuristics of Cohen, Lenstra, Martinet, Malle and others predict the distribution of the p-part of the class group in families of number fields with prescribed degree and Galois group G, at least when p does not divide the order of G. Results in this direction are few and far between, most notably the results of Davenport--Heilbronn and Bhargava. We give a new data point by computing the average size of the 2-torsion in the class group of the cubic fields Q(n^{1/3}). Along the way, we prove a "reciprocity law" for 2-torsion ideal classes in general cubic fields, which indicates how the existing heuristics should be adjusted when p = 2. This is joint work with Artane Siad.

Abstract: When you want to solve a Diophantine equation in practice, one of the most powerful modern techniques is the Chabauty—Kim method, which proceeds by constructing p-adic functions vanishing on the rational solutions of your equation. In this talk, I will discuss a heuristic which predicts when these functions cut out exactly the set of rational points. As we will see, the naive version of the heuristic is not quite correct, since there can occasionally be extra "unexpected" algebraic irrational points where the functions vanish (in accordance with a conjecture of Stoll). I will outline what is known about these unexpected points, including some ongoing work with Jennifer Balakrishnan.

Abstract: We'll begin by looking at the lacunary zeta function $\sum 1/F(n)^s$, where $F(n)$ is the $n$th Fibonacci number. Surprisingly (at least to me), this is deeply connected to modular forms, and a small generalization is connected to counting 3-term arithmetic progressions of squares. This includes work with Eran Assaf, Chan Ieong Kuan, Thomas Hulse, Alexander Walker, and Raphael Steiner.
 

The rational points on an elliptic curve ($y^2 = x^3 + Ax + B$) form a finitely generated abelian group. Heuristics imply that 50% of elliptic curves have rank 0 and 50% have rank 1. For example, this follows from Poonen-Rains' heuristics for the dimensions of the p-Selmer groups of elliptic curves, for all primes p. Bhargava-Shankar computed the first moment of these distributions for p = 2,3,5 and recently Bhargava-Shankar-Swaminathan gave an upper bound on the second moment when p = 2. I'll give a leisurely explanation of the definitions and the methods, and then describe work in progress with Bhargava-Ho-Swaminathan where we prove something close to the full Poonen-Rains distribution for p = 2 in certain large families of elliptic curves.  

We discuss work in progress with Joseph Leung on power-saving estimates for sums of (GL(3) and GL(2)) divisor function and automorphic form coefficients along binary cubic and quartic forms, respectively.

The Bernstein center of a $p$-adic group is a commutative ring of certain distributions on the group, and it interacts closely with the group’s representation theory. Fargues and Scholze provide an abstract construction of a class of elements of the Bernstein center called excursion operators, which encode a candidate for the (semisimplified) local Langlands correspondence. In this talk, I will present an approach to understanding excursion operators concretely as distributions on the group, with a special emphasis on the case of $G = \mathrm{SL}_2$ where everything can be made quite explicit.