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.
Souparna Purohit, University of Pennsylvania
Given an arithmetic variety $\mathscr{X}$ and a hermitian line bundle $\overline{\mathscr{L}}$, the arithmetic Hilbert-Samuel theorem describes the asymptotic behavior of the co-volumes of the lattices $H^0(\mathscr{X}, \mathscr{L}^{\otimes k})$ in the normed spaces $H^0(\mathscr{X}, \mathscr{L}^{\otimes k}) \otimes \mathbb{R}$ as $k \to \infty$. 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 $\overline{\mathscr{L}}$ 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 $\Gamma \subseteq \text{PSL}_2(\mathbb{Z})$ endowed with the logarithmically singular Petersson metric. This generalizes work of Chinburg, Guignard, and Soul\'{e} addressing the case $\Gamma = \text{PSL}_2(\mathbb{Z})$.
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.
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.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.
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.
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.
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.
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.
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.
Preston Wake, Michigan State University
For a prime number N, Ogg's conjecture states that the torsion in the Jacobian of the modular curve $X_0(N)$ 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.