Philadelphia Area Number Theory Seminar 2024
Current contacts: Jaclyn Lang, Catherine Hsu, Ian Whitehead, and Djordje Milicevic
The Philadelphia Area Number Theory Seminar rotates between Bryn Mawr, Swarthmore, and Temple. In Fall 2022, we meet on Tuesday afternoons, usually with tea at 3pm and then the talk 3:30-5pm. Please pay careful attention to the times and locations of the talks as they change from week to week! If you would like to be added to our mailing list or if you are interested in being a speaker, please contact one of the organizers. In future semesters, we anticipate that the seminar will be on Wednesday afternoons.
Click on title for abstract.
Wednesday February 7, 2024 at 14:30, Temple University, Wachman Hall 413
Distribution of the successive minima of the Petersson norm on cusp forms
Souparna Purohit, University of Pennsylvania
Given an arithmetic variety 𝒳 and a hermitian line bundle ℒ⎯⎯⎯⎯⎯⎯, the arithmetic Hilbert-Samuel theorem describes the asymptotic behavior of the co-volumes of the lattices H0(𝒳,ℒ⊗k) in the normed spaces H0(𝒳,ℒ⊗k)⊗ℝ as k→∞. Using his work on quasi-filtered graded algebras, Chen proved a variant of the arithmetic Hilbert-Samuel theorem which studies the asymptotic behavior of the successive minima of the lattices above. Chen's theorem, however, requires that the metric on ℒ⎯⎯⎯⎯⎯⎯ is continuous, and hence does not apply to automorphic vector bundles for which the natural metrics are often singular. In this talk, we discuss a version of Chen's theorem for the line bundle of modular forms for a finite index subgroup Γ⊆PSL2(ℤ) endowed with the logarithmically singular Petersson metric. This generalizes work of Chinburg, Guignard, and Soul\'{e} addressing the case Γ=PSL2(ℤ).Wednesday February 21, 2024 at 14:30, Temple University, Wachman Hall 413
The Schottky problem in characteristic p
Steven Groen, Lehigh University
The Schottky problem is a classical problem that asks which Abelian varieties are isomorphic to the Jacobian of a (smooth) curve. If the dimension exceeds 3, not every Abelian variety can be a Jacobian. In characteristic p, there are additional tools that shed light on this question. In particular, the Ekedahl-Oort stratification partitions Abelian varieties by their p-torsion group scheme. An example of this is the distinction between ordinary elliptic curves and supersingular elliptic curves. The Ekedahl-Oort stratification leads to the following question: which p-torsion group schemes arise from Jacobians of (smooth) curves? Although this question is still wide open, I will present some progress on it, in particular when the curves in question are Artin-Schreier covers. Part of this is joint work with Huy Dang.Wednesday February 28, 2024 at 15:30, Bryn Mawr College, Park Science Center 328
On the Density Hypothesis for Families of Lattices
Djordje Milicevic, Bryn Mawr College
Selberg’s celebrated Eigenvalue Conjecture states that all nonzero Lapla- cian eigenvalues on congruence quotients of the upper half-plane are at least 41 . This particularly strong form of the “spectral gap” property can be thought of as the archimedean counterpart of the Ramanujan–Petersson conjecture for Hecke eigen- values of cusp forms, is expected to suitably hold for more general Lie groups and their arithmetic quotients, and remains far from resolution.
For analytic applications in a family of automorphic forms, in the absence of Selberg’s conjecture, the non-tempered spectrum can often be satisfactorily handled if the exceptions in the family are known to be “sparse” and “not too bad”, in a sense made precise by the so-called density hypothesis evoking the classical density estimates of prime number theory.
In this friendly talk, we will first talk about the density hypothesis in general and how one can go about proving such an estimate. We will then present our recent result (joint with Fra ̧czyk, Harcos, and Maga) establishing the density hypothesis for a broad natural “horizontal” family of not necessarily commensurable arithmetic orbifolds, with uniform power-saving estimates in the volume and spectral aspects.
Wednesday March 20, 2024 at 14:30, Temple University, Wachman Hall 413
Upper and lower bounds for the large deviations of Selberg's central limit theorem
Emma Bailey, CUNY
Suppose we form a complex random variable by evaluating the Riemann zeta function at a random uniform height on the critical line, 1/2 + i U. Selberg’s central limit theorem informs us that the real (or indeed the imaginary) part of the logarithm of this random variable behaves, as T grows, like a centred Gaussian with a particular variance. It is of interest, in particular in relation to the moments of the Riemann zeta function, to understand the large deviations of this random variable. In this talk I will discuss the case for the right tail, presenting upper and lower bounds in work joint with L-P Arguin.Monday April 1, 2024 at 16:00, Temple University, Tuttleman Learning Center, 101
Archimedes' giant troll
Akshay Venkatesh, Institute for Advanced Study
A 22-line poem written around 200 B.C. and attributed to Archimedes challenges the diligent and wise recipient to compute the number of cattle in the herds of the Sun God, after enumerating various properties of these herds. In mathematical language, the lines of the poem translate into a system of constraints amounting to a quadratic equation in two variables; but the resulting question required a further 2000 years to solve. (It turns out that the Sun God has no shortage of cows.) I will tell the story of this puzzle and its underlying mathematics, which is amazingly rich, inspiring mathematicians from Ancient Greece and medieval India to the present.
This lecture is intended for a general audience.
Wednesday April 3, 2024 at 11:00, Temple University, Wachman Hall 617
The duality paradigm for compact groups
Akshay Venkatesh, Institute for Advanced Study
It is a remarkable fact that there is a duality on the set of compact connected Lie groups; this duality interchanges, for example, the rotation group in three dimensions, and the group of unitary two-by-two matrices with determinant 1. This duality emerged in mathematics in the 1960s and, independently, in physics in the 1970s. In mathematics, it has served as an organizing principle for a great variety of phenomena related to Lie group theory, much of which falls under the heading of the “Langlands program”. I will describe some of the history and then two more recent developments: the realization that the mathematical and physics contexts for the duality are actually related to one another, and my recent work with Ben-Zvi and Sakellaridis where we seek to incorporate into the duality spaces upon which the group acts.
Wednesday April 3, 2024 at 14:00, Temple University, Wachman Hall 617
The AI mathematician
Akshay Venkatesh, Institute for Advanced Study
It is likely that developments in automated reasoning will transform research mathematics. I will discuss some ways in which we mathematicians might think about and approach this. The talk will *not* be about current or potential abilities of computers to do mathematics — rather I will look at topics such as the history of automation and mathematics, and related philosophical questions.Wednesday April 10, 2024 at 15:00, Swarthmore College, Science Center Room 181
A refined random matrix model for function field L-functions
Will Sawin, Columbia University
The moments of the absolute value of the Riemann zeta function, up to height T, are expected to be a certain polynomial in log T. Since Keating and Snaith, the random matrix model for the Riemann zeta function has been used not just to model the distribution of its zeroes but the distribution of its values as well, which should include the moments. A modified random matrix model due to Gonek, Hughes, and Keating predicts the leading term of the moment polynomial but not the lower-order terms. In, for now, the function field case, I propose a different modification of the random matrix model. In work in progress, I show this model predicts all terms of the moment polynomial when q is sufficiently large.Wednesday April 17, 2024 at 15:30, Bryn Mawr College, Park Science Center 245
Arithmetic and Topology of Modular Knots
Christopher-Lloyd Simon, Pennsylvania State University
We study several arithmetic and topological structures on the set of conjugacy classes of the modular group PSL(2;Z), such as equivalence relations or bilinear functions.
A) The modular group PSL(2; Z) acts on the hyperbolic plane with quotient the modular orbifold M, whose oriented closed geodesics correspond to the hyperbolic conjugacy classes in PSL(2; Z). For a field K containing Q, two matrices of PSL(2; Z) are said to be K-equivalent if they are conjugated by an element of PSL(2;K). For K = C this amounts to grouping modular geodesics of the same length. For K = Q we obtain a refinement of this equivalence relation which we will relate to genus- equivalence of binary quadratic forms, and we will give a geometrical interpretation in terms of the modular geodesics (angles at the intersection points and lengths of the ortho-geodesics).
T) The unit tangent bundle U of the modular orbifold M is a 3-dimensional manifold homeomorphic to the complement of trefoil in the sphere. The modular knots in U are the periodic orbits for the geodesic flow, lifts of the closed oriented geodesics in M , and also correspond to the hyperbolic conjugacy classes in PSL(2; Z). Their linking number with the trefoil is well understood as it has been identified by E. Ghys with the Rademacher cocycle. We are interested in the linking numbers between two modular knots. We will show that the linking number with a modular knot minus that with its inverse yields a quasicharacter on the modular group, and how to extract a free basis out of these. For this we prove that the linking pairing is non degenerate. We will also associate to a pair of modular knots a function defined on the character variety of PSL(2;Z), whose limit at the boundary recovers their linking number.
Wednesday April 24, 2024 at 14:30, Temple University, Wachman Hall 413
Rational torsion in modular Jacobians
Preston Wake, Michigan State University
For a prime number N, Ogg's conjecture states that the torsion in the Jacobian of the modular curve X0(N) is generated by the cusps. Mazur proved Ogg's conjecture as one of the main theorems in his "Eisenstein ideal" paper. I'll talk about a generalization of Ogg's conjecture for squarefree N and a proof using the Eisenstein ideal. This is joint work with Ken Ribet.Wednesday September 11, 2024 at 14:45, Swarthmore College Science Center 149
Preview of Lectures 1 & 2 in mini-course on local fields
Catherine Hsu, Swarthmore College
In these lectures, we explore the theory of discrete valuations, including both arithmetic and topological properties. We conclude by defining non-archimedean local fields.
Wednesday September 18, 2024 at 14:30, Temple University, Wachman Hall 413
The geometry and arithmetic of the Ceresa cycle
Ari Shnidman, IAS
An algebraic curve C of genus g > 1 can be embedded in its Jacobian variety J in two different ways: {x - e : x in C} and {e - x : x in C}. The Ceresa cycle k(C) is the formal difference of these two curves-- it vanishes if and only if these two curves can be algebraically deformed to each other within J. To date, the only known cases of k(C) = 0 are hyperelliptic curves. However, many examples of k(C) being torsion have recently been found, among curves/Jacobians with symmetries. I'll survey these recent examples, with an emphasis on what number theory has to say about this (a priori) purely algebro-geometric question.Wednesday September 25, 2024 at 14:30, Temple University, Wachman Hall 413
Convergent Decomposition Groups and an S-adic Shafarevich Conjecture
Andrew Kwon, UPenn
In the first part of this talk, I will survey some of the landmark results that demonstrate how closely intertwined arithmetic, Galois theory, and model theory can be (especially situations when "Galois theory = arithmetic" and "model theory = arithmetic" for nice fields). Special attention will be given to decomposition groups and local-global principles, which are the main characters for the second half.
In the second part of the talk, I will discuss known and new results about decomposition groups that generate (generalized) profinite free products and how this relates to a certain Shafarevich Conjecture. Time permitting, I will also discuss future directions pertaining to reconstruction theorems and decidability.Wednesday October 2, 2024 at 14:45, Swarthmore College Science Center 149
Preview of Lectures 3 & 4 in PAWS mini-course on local fields
Catherine Hsu, Swarthmore College
In these lectures, we will state and prove Hensel's lemma and explore some applications of this important tool. We'll then give a classification of non-archimedean local fields and describe the structure of the multiplicative group of p-adic number fields.Wednesday October 9, 2024 at 14:45, Swarthmore College Science Center 149
Experiments with L-functions
Nathan Ryan, Bucknell College
L-functions are central to many questions in modern number theory in the sense that they encode a lot of information about complex number theoretic objects. While L-functions can be attached to primes, to primes in arithmetic progressions, to modular forms, to elliptic curves, etc., at the end of the day they're just complex analytic functions and they can be evaluated numerically at different points on the complex plane. In this talk, I'll describe two projects that I've worked on. The first is using random matrix theory to model the zeros of L-functions and the second is a method to evaluate L-functions at points in the plane that uses surprisingly little number theoretic information.Wednesday October 23, 2024 at 14:30, Temple University, Wachman Hall 413
Generalised Eigenvalues and Flach classes
Alice Pozzi, University of Bristol
The connection between eigenvalues of Hecke operators acting on spaces of automorphic forms and representations of Galois groups is the central theme of the Langlands program. When the action of the Hecke algebra is not semisimple, one can define the finer notion of generalised Hecke eigenvalues; these turn out to be closely related to deformations of Galois representations. In this talk, I will discuss instances of this phenomenon arising for weight 2 mod p modular forms exploiting certain cohomology classes for the symmetric square representation constructed by Flach. This is joint work with Henri Darmon.Wednesday October 30, 2024 at 15:00, Temple University, Wachman Hall 413
Geometric aspects of general multiple Dirichlet series over function fields
Matthew Case-Liu, Columbia University
Multiple Dirichlet series were originally defined as multi-variate Dirichlet series satisfying certain functional equations with conjecturally nice analytic properties that would give precise asymptotics for moments of L-functions. Generalizing an observation of Chinta, Sawin recently gave an axiomatic characterization of a general class of multiple Dirichlet series over function fields that is independent of their functional equations. Moreover, he proved their existence as formal power series by exhibiting the coefficients as trace functions of explicit perverse sheaves.
In this talk, I'll explain how to 1. prove analyticity of these series in a suitable non-empty region of convergence, and 2. establish some (but not all) of the functional equations that they satisfy. The methods for both are completely geometric: analyticity is a consequence of bounding the cohomology of local systems on a compactification of a configuration space, and the functional equations follow from a density trick for irreducible perverse sheaves.Wednesday November 6, 2024 at 14:45
Preview of Lectures 5 & 6 in mini-course on local fields
Catherine Hsu, Swarthmore College
Wednesday November 20, 2024 at 14:30, Temple University, Wachman Hall 413
Tate classes and automorphic forms: a GSp(4) example
Naomi Sweeting, IAS/Princeton
The Hodge and Tate Conjectures aim to describe the part of cohomology coming from algebraic cycles on an algebraic variety. In this talk, I'll describe some Tate classes on the product of a modular curve and a Siegel 3-fold, that can be constructed using automorphic forms. As it turns out, one can prove that some -- but not all -- of these classes come from an algebraic subvariety with a simple moduli interpretation. One can also prove that all the Tate classes in this family of examples are Hodge classes, a result which is a kind of cohomological shadow of Tate's conjecture.The talk will start with a review of the Hodge and Tate Conjectures, the expected relation between them, and what is already known about the conjectures for Shimura varieties.