# Philadelphia Area Number Theory Seminar

Hosts research talks on a range of number theory and related topics.

## About the Seminar

Topics include analytic and algebraic number theory, modular forms, Diophantine equations, automorphic forms and *L*-functions, and more.

We gather a number of theorists from Philadelphia's and nearby math departments, including Bryn Mawr, CUNY, Drexel, Franklin & Marshall, Haverford, Muhlenberg, Penn, PSU, Rowan, Swarthmore, Temple, Villanova, West Chester, and others. If you are a number theorist visiting Philadelphia, we would love to host your talk; contact any of the organizers to arrange for a specific week.

The seminar is co-organized by Djordje Milićević at Bryn Mawr College, Ian Whitehead and Cathy Hsu at Swarthmore College, and Jackie Lang at Temple University. The seminar was previously organized by Helen Grundman. Until Spring 2014, the seminar met at Temple University; see previous talks there.

The seminar meets on Wednesday afternoons,** **rotating through Bryn Mawr College, Swarthmore College, and Temple University. Tea and cookies will be served before each talk.

## Spring 2023 Seminars

**Ravki** (The University of Pennsylvania)

"Serre curves relative to obstruction modulo 2"

Abstract: Let E be an elliptic curve defned over Q. Fix an algebraic closure Q of Q. We get a Galois representation ρE : Gal(Q/Q) → GL2(Zˆ) associated to E by choosing 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 defned 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 ofer an application to the cyclicity problem. In this way, we generalize some foundational results on Serre curves.

**Liyang Yang** (Princeton University)

"Relative Trace Formula and *L*-functions for GL(*n* + 1) × GL(*n*)"

Abstract: 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 nonvnanishing in the level aspect. Further applications to the subconvexity problem will be discussed if time permits.

**Ingrid Vukusic **(University of Salzberg and Franklin & Marshall College)

"The Cancelling Game"

Abstract: 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 k1 kn a · · · a = 1. 1 n After an unconventional introduction, we will ask many questions related to consecutive 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.

**Valentin Blomer** (University of Bonn, Hausdorff Center for Mathematics)

"Quantum Chaos, Quadratic Forms and Fibonacci Numbers"

**Joint Talk with Bi-Co Math Colloquium**

Abstract: In this talk we show how elementary and advanced number theory can be used to shed light on a deep question in spectral geometry: what is the eigenvalue distribution of a compact Riemannian manifold? In the simplest case of a flat torus, this has rather surprising arithmetic features and offers a fascinating interplay between number theory and analysis.

Zvi Shem-Tov, Institute for Advanced Study

"Arithmetic Quantum Unique Ergodicity for 3-Dimensional Hyperbolic Manifolds"

Abstract: 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.

**SPRING BREAK - NO SEMINAR**

**Samit Dasgupta** (Duke University)

"Ribet’s Lemma, the Brumer-Stark Conjecture, and the Main Conjecture"

Abstract: In 1976, Ken Ribet used modular techniques to prove an important relationship between class groups of cyclotomic felds 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 felds. 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 fnds 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 felds 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 feld, which has many important corollaries. This is joint work with Mahesh Kakde, Jesse Silliman, and Jiuya Wang.

**Christopher Lutsko** (Rutgers University)

"Fine Scale Properties of Sequences Modulo 1"

Abstract: 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)

"Massey Products and Elliptic Curves"

Abstract: This is joint work with T. Chinburg and J. Gillibert. The application of Massey products to understand the Galois groups of extensions of number felds is a longstanding research topic. In 2014, Minac and Tan showed that triple Massey products vanish for the absolute Galois group of any feld F. In 2019, Harpaz and Wittenberg showed that this remains true for all higher Massey products in the case when F is a number feld. The frst natural case to consider beyond felds is that of Massey products for curves over felds. I will discuss some known and new vanishing and non-vanishing results in this case. In particular, for elliptic curves I will provide a classifcation 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 frst arose in topology, and about the relevant representation theory, before discussing our results.

**Rahul Dalal **(Johns Hopkins University)

"Applications of the Endoscopic Classification to Statistics of Cohomological Automorphic Representations on Unitary Groups"

Abstract: Starting from the example of classical modular modular forms, we motivate and describe the problem of computing statistics of automorphic representations. 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 nontempered) cohomological representation π0 at infinity and level dividing some finite upper bound. We compute statistics of this family as the level restriction goes to infinity. 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 corollaries 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 representations 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)

"The Nonvanishing of Selmer Groups for Certain Symplectic Galois Representations"

Abstract: Given an automorphic representation π of SO(n, n + 1) with certain nice properties at infnity, 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 ramifed

**Kalyani Kansal** (Johns Hopkins University)

"Intersections of Components of Emerton-Gee stack for GL2"

Abstract: The Emerton-Gee stack for GL2 is a stack of (φ, Γ)-modules of rank two. Its reduced part, X, is an algebraic stack of fnite type over a fnite feld, 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.

## Fall 2022 Seminars

**Manami Roy **(Fordham University)

"Dimensions of Spaces of Siegel Cusp Forms of Degree 2"

Abstract: Computing dimension formulas for the spaces of Siegel modular forms of degree 2 is of great interest to many mathematicians. We will start by discussing known results and methods in this context. The dimensions of the spaces of Siegel cusp forms of non-squarefree levels are mostly unavailable in the literature. This talk will present new dimension formulas of Siegel cusp forms of degree 2, weight k, and level 4 for three congruence subgroups. Our method relies on counting a particular set of cuspidal automorphic representations of GSp(4) and exploring its connection to dimensions of spaces of Siegel cusp forms of degree 2. This work is joint with Ralf Schmidt and Shaoyun Yi.

**Luochen Zhao** (Johns Hopkins University)

"Explicit period formulas for totally real p-adic L-functions, a la Cassou-Nogues"

Abstract: The p-adic Hecke L-functions over totally real fields are known to exist by works of Deligne-Ribet, Cassou-Nogu´es and Barsky in the late 70s, albeit the whole picture of which is still clouded to this day. In this talk I will report my recent work on the explict determination of the incarnate p-adic measures that generalizes the p-adic Bernoulli distributions, and its applications in the Gross-Stark conjecture and totally real Iwasawa invariants.

**Djordje Milićević** (Bryn Mawr College)

"Nonvanishing of Dirichlet L-functions"

Abstract: Central values of L-functions encode essential arithmetic information. A host of theorems and widely believed conjectures predict that they should not vanish unless there is a deep arithmetic reason for them to do so (and that this should be an exceptional occurrence in suitably generic families). In particular, it is conjectured that L( 1 2 , χ) ≠ 0 for every Dirichlet character χ. In this talk, I will begin with a non-technical overview of the analytic numbertheoretic techniques used to establish non-vanishing of L-functions and then present recent progress, in joint work with Khan and Ngo, on the non-vanishing problem for Dirichlet L-functions to large prime moduli, which also leverages deep estimates on exponential sums.

**Louis Gaudet** (Rutgers University)

"The least Euler prime via a sieve approach"

Abstract: Euler primes are primes of the form p = x2 + Dy2 with D > 0. In analogy with Linnik’s theorem, we can ask if it is possible to show that p(D), the least prime of this form, satisfies p(D) DA for some constant A > 0. Indeed Fogels showed this in 1962, but it wasn’t until 2016 that an explicit value for A was determined by Zaman and Thorner, who showed one can take A = 694. Their work follows the same outline as the traditional approach to proving Linnik’s theorem, relying on log-free zero-density estimates for Hecke L-functions and a quantitative Deuring–Heilbronn phenomenon. In an ongoing work (as part of my PhD thesis) we propose an alternative approach to the problem via sieve methods that avoids the use of the above technical results on zeros of the Hecke L-functions. We hope that such simplifications may result in a better value for the exponent A.

**Renee Bell** (CUNY Lehman)

"How Do Points on Plane Curves Generate Fields? Let Me Count the Ways."

Abstract: In their program on diophantine stability, Mazur and Rubin suggest studying a curve C over Q by understanding the field extensions of generated by a single point of C; in particular, they ask to what extent the set of such field extensions determines the curve. A natural question in arithmetic statistics along these lines concerns the size of this set: for a smooth projective curve C how many field extensions of Q — of given degree and bounded discriminant — arise from adjoining a point of C? Can we further count the number of such extensions with specified Galois group? Asymptotic lower bounds for these quantities have been found for elliptic curves by Lemke Oliver and Thorne, for hyperelliptic curves by Keyes, and for superelliptic curves by Beneish and Keyes. We discuss similar asymptotic lower bounds that hold for all smooth plane curves C, using tools such as geometry of numbers, Hilbert irreducibility, Newton polygons, and linear optimization.

**Catherine Hsu** (Swarthmore College)

"Explicit Non-Gorenstein R = T via Rank Bounds"

Abstract: In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk, we’ll explore generalizations of Mazur’s result to squarefree level, focusing on recent work, joint with P. Wake and C. Wang-Erickson, about a non-optimal level N that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein. First, we will outline a Galois-theoretic criterion for the deformation ring to be as small as possible, and when this criterion is satisfied, deduce an R = T theorem. Then we’ll discuss some of the techniques required to computationally verify the criterion.

**FALL BREAK - NO SEMINAR**

**Ellen Eischen** (University of Oregon)

*"p*-adic Aspects of Modular Forms and L-functions"

Abstract: I will discuss recent developments and ongoing work for p-adic aspects of modular forms and L-functions, which encode arithmetic data. Interest in p-adic properties of values of L-functions originated with Kummer’s study of congruences between values of the Riemann zeta function at negative odd integers, as part of his attempt to understand class numbers of cyclotomic extensions. After presenting an approach to proving congruences and constructing p-adic L-functions, I will conclude the talk by introducing ongoing joint work of G. Rosso, S. Shah, and myself (concerning Spin L-functions for GSp 6). I will explain how this work fit into the context of earlier developments, including constructions of Serre, Katz, Coates–Sinnot, Deligne–Ribet, Hida, E–Harris– Li–Skinner, and Liu. I will not assume the audience has prior familiarity with p-adic L-functions or Spin L-functions, and all who are curious about this topic are welcome

**Amanda Folsom** (Amherst College)

"Symmetry, almost"

Some definitions of the word symmetry include “correct or pleasing proportion of the parts of a thing,” “balanced proportions,” and “the property of remaining invariant under certain changes, as of orientation in space.” One might think of snowflakes, butterflies, and our own faces as naturally symmetric objects – or at least close to it. Mathematically one can also conjure up many symmetric objects: even and odd functions, fractals, certain matrices, and modular forms, a type of symmetric complex function. All of these things, mathematical or natural, arguably exhibit a kind of beauty in their symmetries, so would they lose some of their innate beauty if their symmetries were altered? Alternatively, could some measure of beauty be gained with slight symmetric imperfections? We will explore these questions from past to present guided by the topic of modular forms and their variants. What can be gained by perturbing modular symmetries in particular?

**Debanjana Kundu** (University of British Columbia)

"Heuristics for Anti-cyclotomic Zp-extensions"

Abstract: For an imaginary quadratic field, there are two natural Zp-extensions, the cyclotomic and the anticyclotomic. We’ll start with a brief description of Iwasawa theory for the cyclotomic extensions, and then describe some computations for anticyclotomic Zp extensions, especially the fields and their class numbers. This is joint work with LC Washington.

**Louis-Pierre Arguin** (Baruch College, CUNY)

"Large Values of the Riemann zeta function on the Critical Line"

Abstract: The interplay between probability theory and number theory has a rich history of producing deep results and conjectures. This talk will review recent results in this spirit where the insights of probability have led to a better understanding of large values of the Riemann zeta function on the critical line. In particular, we will discuss the large deviations of Selberg’s central limit theorem as well as the maximum of zeta in short intervals. This is based on joint works with Emma Bailey, and with Paul Bourgade & Maksym Radziwill.

## Contact Us

Mathematics Department

**Tina Fasbinder**

Academic Administrative Assistant

tfasbinder@brynmawr.edu

610-526-5348

Fax: 610-526-6575

Park Science Building

Bryn Mawr College

Bryn Mawr, PA 19010-2899