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.
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.
The p-adic Hecke L-functions over totally real fields are known to exist by works of Deligne-Ribet, Cassou-Nogues 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.
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 number- theoretic 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
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.
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.
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.
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.
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?
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.
Originally studied by Mazur in the early 1990s, the deformation theory of Galois representations describes the ways in which one can lift a mod p Galois representation to characteristic zero. It plays a central role in the Langlands program; for instance, a careful study of the geometry of deformation rings is one of the key inputs to Wiles’s proof of Fermat’s last theorem.
I will give a short introduction to deformation theory in general, and then explain how Galois representations fit into this framework. Then I will give a sketch of how these are used in modularity lifting theorems. Finally, I will talk about work with Gebhard Böckle and Vytautas Paškūnas which describes the geometry of local p-adic Galois deformation rings. If time permits I will discuss the strategy of the proof, which involves a reduction to the theory of pseudorepresentations.
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.