Bay Area Algebraic Number Theory and Arithmetic Geometry Day 15

Saturday, December 2, 2017
University of California, Berkeley
Rooms: Evans Hall room 740 for lectures, 1015 for breaks


Speakers:

Julia Gordon (UBC)
Jeremy Lovejoy (CNRS/Berkeley)
Yiannis Sakellaridis (Rutgers)
Jan Vonk (McGill)
Preston Wake (UCLA)


Schedule:

9:30-10:00 Coffee/Bagels (Evans 1015)
10:00-11:00 Julia Gordon
11:00-11:15 Coffee Break
11:15-12:15
Jeremy Lovejoy
12:15-1:45
Lunch
1:45-2:45 Yannis Sakellaridis
2:45-3:10
Coffee Break
3:10-4:10
Jan Vonk
4:10-4:30
Break
4:30-5:30
Preston Wake
6:00
Dinner, location TBD
Please RSVP to sdasgup2 (at) ucsc (dot) edu





Titles and Abstracts:

Julia Gordon, "Measures, orbital integrals, and product formulas for isogeny classes of abelian varieties"

In 2003, E.-U. Gekeler gave a formula for the number of elliptic curves in an isogeny class (over a finite field), based on probabilistic and equidistribution considerations. This formula is in a sense similar to Siegel's formula expressing the size of a genus of a quadratic form as a product of local densities. It is well-known that Siegel's formula is equivalent to the computation of the Tamagawa volume of the orthogonal group. In the same spirit, there is a formula, due to Langlands and Kottwitz, expressing the cardinality of an isogeny class of principally polarized abelian varieties as an adelic orbital integral. It turns out that by carefully comparing different natural measures on the orbits, one can see a direct connection between Gekeler's formula and the formula of Langlands and Kottwitz, and therefore generalize Gekeler-style formula to higher dimension. In the process we encounter some questions that appear classical but seem to be surprisingly difficult to answer, which I am hoping to discuss. This is a joint project with Jeffrey Achter, Salim Ali Altug, and Luis Garcia.

Jeremy Lovejoy, "Colored Jones polynomials and modular forms"

In this talk I will discuss joint work with Kazuhiro Hikami, in which we use Bailey pairs and the Rosso-Jones formula to compute the cyclotomic expansion of the colored Jones polynomial of a certain family of torus knots. As an application we find quantum modular forms dual to the generalized Kontsevich-Zagier series. As another application we obtain formulas for the WRT invariants of certain 3-manifolds, some of which reveal mock modularity. I will also touch on joint work with Robert Osburn, in which we compute a formula for the colored Jones polynomial of double twist knots.

Yiannis Sakellaridis, "Transfer operators between relative trace formulas in rank one"

I will introduce a new paradigm for comparing relative trace formulas, in order to prove instances of (relative) functoriality and relations between periods of automorphic forms. More precisely, for a spherical variety X=H\G of rank one, I will prove that there is an explicit "transfer operator" which transforms the orbital integrals of the relative trace formula for X x X/G to the orbital integrals of the Kuznetsov formula for GL(2) or SL(2), equipped with suitable non-standard test functions. The operator is determined by the L-value associated to the square of the H-period integral, and the proof uses a deep theory of Friedrich Knop on the cotangent bundles of spherical varieties. This is part of an ongoing joint project with Daniel Johnstone and Rahul Krishna, who are proving instances of the fundamental lemma. Globally, this transfer will induce an identity of relative trace formulas and global relative characters, translating to an Ichino–Ikeda type formula that relates the square of the H-period to the said L-value. This can be viewed as part of the program of relative functoriality, a generalization of the Langlands functoriality conjecture, predicting relations between the automorphic spectra of two spherical varieties when there is a map between their dual groups. The case under consideration here is the simplest non-abelian case of this, when the dual groups are equal and of rank one. If time permits, I will discuss how the transfer operator here and in a few examples of higher rank where it is known is a "deformation" of an abelian transfer operator obtained by replacing the spherical variety by its asymptotic cone (or boundary degeneration).

Jan Vonk, "Singular moduli for real quadratic fields"

The theory of complex multiplication describes finite abelian extensions of imaginary quadratic number fields using singular moduli, which are special values of modular functions at CM points. Hilbert's 12th problem asks for a satisfactory analogue of this theory for arbitrary number fields. I will describe joint work with Henri Darmon in the setting of real quadratic fields, where we construct p-adic analogues of singular moduli through classes of rigid meromorphic cocycles.

Preston Wake, "The rank of Mazur's Eisenstein ideal"

In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We use deformation theory of pseudorepresentations to study the corresponding Hecke algebra. We will discuss how this method can be used to refine Mazur's results, quantifying the number of Eisenstein congruences. Time permitting, we'll also discuss some partial results in the composite-level case. This is joint work with Carl Wang-Erickson.


Parking:

To park in a campus parking lot, you can purchase a permit from a ticket dispenser located in the lot. Permits should be placed on your dashboard. It costs $12 to park all day, and the dispensers take credit cards. If you have a parking permit of the correct "strength" from another UC campus, you can park for free by displaying your UC permit.

The Berkeley campus has a method to pay for campus parking via mobile phones. To use this service, you can set up an account ahead of time as described on the web page and download the iPhone or Android app ahead of time as well.

A list of parking lots is available here; clicking on the fifth link "Campus Parking Lots" sends you to a Google map displaying the lots. Scroll down on the left panel until you see the name of the desired parking lot; click on the name and you get a description of the lot and its capacities.

Two recommended parking locations are the Upper Hearst Parking Structure and the Lower Hearst Parking Structure. The easiest solution is to park on level 1 of the Lower Hearst Structure. This structure is on the north side of Hearst Avenue, just south of Euclid. Because Hearst is a divided road, you need to be heading west on Hearst in order to turn into the lot. Walking to Evans Hall from either structure takes less than 5 minutes.


Registration:

There is no formal registration, but if you plan to attend, we would appreciate an email to sdasgup2 at ucsc dot edu to help plan the event, especially if you plan to attend the dinner afterwards.


Dinner:

There will be a dinner following the conference at 6:00pm, at a location to be announced. Please send an email to sdasgup2 at ucsc dot edu if you plan to attend. Graduate students will be partially subsidized at the dinner. We thank the UC Santa Cruz Mathematics Department, the Math Research Center at Stanford University, and the UC Berkeley Mathematics Department for partial financial support.