Collaborative Number Theory Seminar at the CUNY Graduate Center
Co-organizers: Gautam Chinta, Brooke Feigon, Krzysztof Klosin,
Fall 2013 Schedule:
September 27: Cormac O'Sullivan (CUNY Bronx Community College)
Title: The partial fraction decomposition of the restricted partition generating function and Rademacher's conjecture
Abstract: The generating function for the partitions of an integer m into at most N parts may be written as a simple product. Rademacher studied the coefficients of the partial fraction decomposition of this product and made a conjecture in 1973 for the limits of these coefficients as N goes to infinity based on his famous exact formula for the unrestricted partitions. This talk describes the latest results on Rademacher's conjecture and connections with the zeros of the dilogarithm function.
October 4: Jeff Breeding (Fordham University)
Constructing Siegel modular forms with Borcherds
Abstract: In this talk, we will describe an exponential lifting of weakly holomorphic Jacobi forms of weight zero into spaces of Siegel modular forms with respect to the paramodular group. We will finish with a presentation of recent progress in constructing a variant of this lift with respect to theta groups.
October 25: John Miller (Rutgers University)
Title: Calculating class numbers beyond Odlyzko's bounds: Real cyclotomic fields
One of the most intensively studied classes of number
fields are the cyclotomic fields. Surprisingly,
their class numbers have only been determined for
fields of rather small conductor, due to the
difficulty of finding the "plus part" of the class
number, i.e. the class number of the maximal real
subfield. For example, the class numbers of the
real cyclotomic fields of prime conductor have only
been determined for primes up to 67.
November 8: Ana Caraiani (Princeton/IAS)
Title: Patching and p-adic local Langlands
The p-adic local Langlands correspondence is well
understood for GL_2(Q_p), but appears much more
complicated when considering GL_n(F), where either
n>2 or F is a finite extension of Q_p. I will
discuss joint work with Matthew Emerton, Toby Gee,
David Geraghty, Vytautas Paskunas and Sug Woo Shin, in
which we approach the p-adic local Langlands
correspondence for GL_n(F) using global methods. The
key ingredient is Taylor-Wiles-Kisin patching of
completed cohomology. This allows us to prove many new
cases of the Breuil-Schneider conjecture.
November 22: Brian Smithling (Johns Hopkins University)
Title: Spin conditions for
Abstract: Local models are
schemes which are intended to model the étale-local
structure of integral models of Shimura varieties.
Pappas and Zhu have recently given a general
group-theoretic definition of local models with
parahoric level structure, valid for any tamely
ramified group, but it remains an interesting problem
to characterize the local models, when possible, in
terms of an explicit moduli problem. In the case
of split GO(2g), Pappas and Rapoport have given a
conjectural moduli description of the local model, the
crucial new ingredient being what they call the _spin
condition_. I will report on the proof of their
conjecture in the case of a certain maximal (but not
hyperspecial) parahoric level. Time permitting,
I will also comment on the case of local models for
ramified, quasi-split unitary groups. Here
Pappas and Rapoport have also introduced a variant of
the spin condition, but it turns out that this needs
to be strengthened.
December 13: Michael Woodbury (Columbia University)
Title: Exceptional Dual Pair Correspondences and Functoriality
Abstract: I will discuss certain dual pairs G x G' occurring in the exceptional groups E_8, E_7, E_6, (and D_5) over a p-adic field. By restricting the minimal representation of the larger group to the dual pair, one obtains a so-called theta correspondence between the representations of G and G’. The use of Jacquet functors gives a way of building up an "inductive” process to prove that, for spherical representations, the theta correspondence is functorial. This is joint work with Gordan Savin.