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.
Rakvi, University of Pennsylvania
Let 𝐸 be an elliptic curve defined over Q. Fix an algebraic closure Q of Q. We get a Galois representation 𝜌𝐸 : Gal(Q/Q) → GL2(Z) associated to 𝐸 by choosing a compatible bases for the 𝑁 -torsion subgroups of 𝐸 (Q). In this talk, I will discuss my recent work joint with Jacob Mayle where we consider elliptic curves 𝐸 defined over Q for which the image of the adelic Galois representation 𝜌𝐸 is as large as possible given a constraint on the image modulo 2. For such curves, we give a characterization in terms of their l-adic images, compute all examples of conductor at most 500,000, precisely describe the image of 𝜌𝐸 , and offer an application to the cyclicity problem. In this way, we generalize some foundational results on Serre curves.
Liyang Yang, Princeton University
We will introduce a relative trace formula on GL(n + 1) weighted by cusp forms on GL(n) over number fields. The spectral side is an average of Rankin– Selberg L-functions for GL(n + 1) × GL(n) over the full spectrum, and the geometric side consists of Rankin–Selberg L-functions for GL(n) × GL(n), and certain explicit meromorphic functions. The formula yields new results towards central L-values for GL(n + 1) × GL(n): the second moment evaluation, and simultaneous nonvnanish- ing in the level aspect. Further applications to the subconvexity problem will be discussed if time permits.
Ingrid Vukusic, University of Salzburg
An n-tuple of integers (a1 , . . . , an ) is called multiplicatively dependent, if it allows you to win the “Cancelling Game”, i.e. if there exist integers k1, . . . , kn ∈ Z, not all zero, such that
a^{k1} ···a^{kn} =1.
After an unconventional introduction, we will ask many questions related to con- secutive tuples of multiplicatively dependent integers, and answer some of them. For example, do there exist integers 1 < a < b such that (a,b) and (a+1,b+1) are both multiplicatively dependent? It turns out that this question is easily answered, and after briefly discussing some more general properties of pairs, we will move on to triples. The proof of the main result relies on lower bounds for linear forms in logarithms. This talk is based on joint work with Volker Ziegler, as well as some work in progress.Zvi Shgem-Tov, IAS
The Quantum Unique Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the Laplacian on a compact manifold of negative curvature become equidistributed as the eigenvalue tends to infinity. In the talk I will discuss a recent work on this problem for arithmetic quotients of the three dimensional hyperbolic space. I will present a rather detailed proof of our key result that these eigenfunctions cannot concentrate on certain proper submanifolds. Joint work with Lior Silberman.
Samit Dasgupta, Duke University
In 1976, Ken Ribet used modular techniques to prove an important relationship between class groups of cyclotomic fields and special values of the zeta function. Ribet’s method was generalized to prove the Iwasawa Main Conjecture for odd primes p by Mazur-Wiles over Q and by Wiles over arbitrary totally real fields.
Central to Ribet’s technique is the construction of a nontrivial extension of one Galois character by another, given a Galois representation satisfying certain properties. Throughout the literature, when working integrally at p, one finds the assumption that the two characters are not congruent mod p. For instance, in Wiles’ proof of the Main Conjecture, it is assumed that p is odd precisely because the relevant characters might be congruent modulo 2, though they are necessarily distinct modulo any odd prime.
In this talk I will present a proof of Ribet’s Lemma in the case that the characters are residually indistinguishable. As arithmetic applications, one obtains a proof of the Iwasawa Main Conjecture for totally real fields at p=2. Moreover, we complete the proof of the Brumer-Stark conjecture by handling the localization at p=2, building on joint work with Mahesh Kakde for odd p. Our results yield the full Equivariant Tamagawa Number conjecture for the minus part of the Tate motive associated to a CM abelian extension of a totally real field, which has many important corollaries.
This is joint work with Mahesh Kakde, Jesse Silliman, and Jiuya Wang.
Christopher Lutsko, Rutgers University
Given a sequence of numbers, a key question one can ask is how is this sequence distributed? In particular, does the sequence exhibit any pseudo-random properties? (i.e., properties shared by random sequences). For example one can ask if the sequence is uniformly distributed modulo 1 (macroscopic scale), or if the pair correlation or gap distribution is Poissonian (fine scale). In this talk I will introduce these concepts, and discuss a set of examples where this behavior is fully understood. The techniques used are common tools in analytic number theory, and the question relates to problems in quantum chaos, and relates to the study of the zeros of the Riemann zeta function (although I will refrain from presenting my proof of RH...). This is joint work with Athanasios Sourmelidis and Nichlas Technau.
Frauke Bleher, University of Iowa
This is joint work with T. Chinburg and J. Gillibert. The application of Massey products to understand the Galois groups of extensions of number fields is a longstanding research topic. In 2014, Minac and Tan showed that triple Massey products vanish for the absolute Galois group of any field F. In 2019, Harpaz and Wittenberg showed that this remains true for all higher Massey products in the case when F is a number field. The first natural case to consider beyond fields is that of Massey products for curves over fields. I will discuss some known and new vanishing and non-vanishing results in this case. In particular, for elliptic curves I will provide a classification for the non-vanishing of triple Massey products under various natural assumptions. The main tool is the representation theory of etale fundamental groups into upper triangular unipotent matrix groups. I will begin with background about Massey products, which first arose in topology, and about the relevant representation theory, before discussing our results.
Rahul Dalal, Johns Hopkins University
Starting from the example of classical modular modular forms, we mo- tivate and describe the problem of computing statistics of automorphic representa- tions. We then describe how techniques using or built off of the Arthur–Selberg trace formula help in studying it.
Finally, we present recent work on one particular example: consider the family of automorphic representations on some unitary group with fixed (possibly non- tempered) cohomological representation π0 at infinity and level dividing some finite upper bound. We compute statistics of this family as the level restriction goes to in- finity. For unramified unitary groups and a large class of π0, we are able to compute the exact leading term for both counts of representations and averages of Satake parameters. We get bounds on our error term similar to previous work by Shin– Templier that studied the case of discrete series at infinity. We also discuss corollar- ies related to the Sarnak–Xue density conjecture, average Sato–Tate equidistribution in families, and growth of cohomology for towers of locally symmetric spaces. The specific new technique making this unitary example tractable is an extension of an inductive argument that was originally developed by Ta ̈ıbi to count unramified rep- resentations on Sp and SO and used the endoscopic classification of representations (which our case requires for non-quasisplit unitary groups).
This is joint work with Mathilde Gerbelli-Gauthier.Sam Mundy, Princeton University
Given an automorphic representation π of SO(n,n+1) with certain nice properties at infinity, one can nowadays attach to π a p-adic Galois representation R of dimension 2n. The Bloch--Kato conjectures then predict in particular that if the L-function of R vanishes at its central value, then the Selmer group attached to a particular twist of R is nontrivial.
I will explain work in progress proving the nonvanishing of these Selmer groups for such representations R, assuming the L-function of R vanishes to odd order at its central value. The proof constructs a nontrivial Selmer class using p-adic deformations of Eisenstein series attached to π, and I will highlight the key new input coming from local representation theory which allows us to check the Selmer conditions for this class at primes for which π is ramified.
Kalyani Kansal, Johns Hopkins University
The Emerton-Gee stack for GL2 is a stack of (phi, Gamma)-modules of rank two. Its reduced part, X, is an algebraic stack of finite type over a finite field, and it can be viewed as a moduli stack of mod p representations of a p-adic Galois group. We compute criteria for codimension one intersections of the irreducible components of X. We interpret these criteria in terms motivated by conjectural categorical p-adic and mod p Langlands correspondence. We also give a cohomological criterion for the number of top-dimensional components in a codimension one intersection.
Xiaoyu (Coco) Huang, CUNY Graduate Center
In this work, we study Fontaine-Laffaille, essentially self-dual deformations of a mod p non-semisimple Galois representation of dimension n with its Jordan-Holder factors being three mutually non-isomorphic absolutely irreducible representations. We show that under some conditions on certain Selmer groups, the universal deformation ring is a discrete valuation ring. Given enough information on the Hecke side, we also prove an R=T theorem. We then apply our results to abelian surfaces with cyclic rational isogenies and certain 6-dimensional representations arising from automorphic forms congruent to Ikeda lifts. In particular, our result identifies the special L-value conditions for the uniqueness of the abelian surface isogeny class, and assuming the Bloch-Kato conjecture, an R=T theorem for the 6-dimensional representations.
Rebecca Bellovin, IAS
The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation \rho:Gal_Q\rightarrow GL_2(\overline{Q_p)$ is modular if it is unramified outside finitely many places and de Rham at p. I will talk about what this means, and I will discuss an analogous modularity result for Galois representations \rho:Gal_Q\rightarrow GL_2(L) when L is instead a non-archimedean local field of characteristic p.
Rakvi, University of Pennsylvania
Let E be a non CM elliptic curve defined over $\mathbb{Q}$. There is an isogeny torsion graph associated to E and there is also a Galois representation $\rho_{E,l} : Gal(\bar{\mathbb{Q}}/\mathbb{Q}) \to GL_2(\mathbb{Z}_l)$ associated to E for every prime l. In this talk, I will discuss a classification of 3 adic Galois images associated to vertices of isogeny torsion graph of E.
Ian Whitehead, Swarthmore College
Recent work of Sawin gives a very general construction of multiple Dirichlet series over function fields satisfying a set of geometric axioms. This construction should encompass all Weyl group multiple Dirichlet series, as well the multiple Dirichlet series associated to higher moments of L-functions and Kac-Moody root systems. The analytic properties of Sawin's multiple Dirichlet series are not yet fully understood. In this talk, I will describe the groups of functional equations satisfied by these series, some proven and some still conjectural. This is joint work in progress with Sawin.
Sean Howe, University of Utah
In classical p-adic geometry, the fundamental objects are rigid analytic spaces built out of convergent power series rings reminiscent of those appearing in complex geometry. These are sufficient for many purposes, but they do not allow for a useful theory of infinite covering spaces. One remedy is to embed rigid analytic spaces in Scholze's category of diamonds by considering the functor of points on perfectoid algebras. The category of diamonds satisfies nice stability properties, including the existence of infinite covering spaces, but these perfectoid test objects behave very differently than the classical convergent power series rings: for example, because perfectoid algebras contain approximate p-power roots, they admit no continuous derivations and thus no tangent space in the classical sense. In this talk, we will survey some interesting phenomena and examples that arise while studying the relation between rigid analytic varieties and more general diamonds, especially in the context of period maps.
Matthew Walsh, University of Maryland
For an integer, monic, irreducible polynomial F, we call the x (mod m) satisfying F(x) ≡ 0 (mod m) the roots of the congruence, and we consider the se- quence of normalized roots x/m ordered by increasing m. For quadratic F , statistical information about this sequence and certain subsequences has proven to be valuable input to many problems in analytic number theory. In joint work with Jens Marklof, we found a dynamical realization of the roots as return times to a specific section for the horocycle flow on SL(2, Z)\SL(2, R), analogous to Athreya and Cheung’s in- terpretation of the BCZ map for Farey fractions. Our realization of the roots leads to limit theorems for the pair correlation and other fine-scale statistics. Similar in- terpretations can be found for cubic and higher degree F but give weaker statistical information than can be obtained in the quadratic setting.
Nina Zubrilina, Princeton University
In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the p-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a much broader class of arithmetic L-functions when split by root number. In my talk, I will discuss this root number correlation bias when the average is taken over all weight k modular newforms. I will point to a source of this phenomenon in this case and compute the correlation function exactly.
Rachel Greenfeld, IAS
Translational tiling is a covering of a space (such as Euclidean space) using translated copies of one building block, called a "translational tile", without any positive measure overlaps. Can we determine whether a given set is a translational tile? Does any translational tile admit a periodic tiling? A well known argument shows that these two questions are closely related. In the talk, we will discuss this relation and present some new developments, joint with Terence Tao, establishing answers to both questions.
Karol Koziol, CUNY Baruch
The classical Satake transform gives an isomorphism between the complex spherical Hecke algebra of a p-adic reductive group G, and the Weyl-invariants of the complex spherical Hecke algebra of a maximal torus of G. This provides a way for understanding the K-invariant vectors in smooth irreducible complex representations of G (where K is a maximal compact subgroup of G), and allows one to construct instances of unramified Langlands correspondences. In this talk, I'll present work in progress with Cédric Pépin in which we attempt to understand the analogous situation with mod p coefficients, and working at the level of the derived category of smooth G-representations.