Collaborative Number Theory Seminar at the CUNY Graduate Center

Co-organizers: Gautam Chinta, Brooke Feigon, Krzysztof Klosin, Maria Sabitova, Lucien Szpiro.

The seminar currently meets Fridays 2:00 to 3:00 PM in Room 3209. The CUNY Graduate Center is located on Fifth Avenue, on the east side of the street, between 34th and 35th Streets in midtown Manhattan. For further information, please contact Maria Sabitova.

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)

Title: Constructing Siegel modular forms with Borcherds products

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

Abstract: 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.

The difficulty is that the Minkowski bound of these fields is very large, and also the root discriminant is too large for the class number to be treated by Odlyzko's discriminant bounds.

Our recent results have improved the situation.  We can now unconditionally prove that the class numbers of the real cyclotomic fields of prime conductor are 1 for primes up to 151.  Furthermore, under the assumption of the generalized Riemann hypothesis, we can calculate the class numbers of real cyclotomic fields up to prime conductor of 241.

This new technique should be applicable to other number fields of moderately large discriminant, allowing us to confront the problem of determining the class number for a large group of number fields which so far have not been treatable by previously known methods.

November 8: Ana Caraiani (Princeton/IAS)

Title: Patching and p-adic local Langlands

Abstract: 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 local models

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.