This semester the seminar meets on Fridays from 4 - 5:30 PM in Room 4422. 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 Gautam Chinta (email@example.com). For information on the seminar in previous years, please see Collaborative Satellite Number Theory Seminar and this link.
Title: Fourier coefficients of Maass forms at various cusps
Abstract: Let f be a newform (Maass form) of level N with a primitive character modulo N. We find the relation between the Fourier coefficients of f at infinity and other cusps using a corresponding automorphic form on the adele group.
Title: Asymptotics for special derivatives of quadratic L-series
Abstract: We shall explain a proof of a quantitative nonvanishing result conjectured by Ph.Michel and A.Venkatesh which concerns the special derivatives occuring in the Gross-Zagier formula. A geometric consequence concerns the rank of rational elliptic curves over Hilbert class fields of imaginary quadratic fields. Then we shall try to summarize in more details what can be said about the height of Heegner points when the discriminant gets large. Both analytic and geometric methods are relevant in this context.
Title: Abelian surfaces of odd conductor: existence, non-existence and modularity
Abstract: The modularity conjecture for two-dimensional abelian varieties A over the rationals would associate to the isogeny class of A a Siegel modular form of weight 2 and specified level. I will discuss joint work with Armand Brumer to test this conjecture. We prove non-existence of certain classes of abelian surfaces by a careful study of their division fields. We also construct abelian surfaces, including some that are not principally polarized. Examples will be given. Our results are compatible with work in progress of Chris Poor and David Yuen on Siegel forms.
Title: Relative trace formulae with a view towards unitary groups
Abstract: Shimura varieties are a class of quasi-projective varieties equipped with rich symmetries due to the action of a large set of correspondences controlled by a reductive group. The study of algebraic cycles on such varieties is of deep arithmetic and geometric interest. In this talk we will explain how the twisted relative trace formula can used to study certain families of cycles on Shimura varieties attached to unitary groups.
Title: Statistics of the zeros of zeta functions in ensembles of hyperelliptic curves over a finite field
Abstract: Much of the Diophantine information concerning curves over a finite field is encoded in the zeta function of the curve. We investigate the statistics of such zeta functions as the curve varies in a moduli space of hyperelliptic curves over a finite field. When the genus g is fixed and the finite field grows, Katz and Sarnak showed that the statistics are those of random unitary symplectic 2g by 2g matrices. I will discuss what is known for the opposite limit, when the finite field is fixed and the genus grows.
Title: Toroidal automorphic forms and the Riemann hypothesis for some function fields of curves
Special time and room: 4pm-5pm in Room 6417. This will be a joint talk with the Commutative Algebra and Algebraic Geometry seminar.
Abstract: In the 1970's, Don Zagier introduced toroidal automorphic
forms to study the zeros of zeta functions. An automorphic form on
GL(2) is toroidal if all its right translates integrate to zero over
all nonsplit tori in GL(2). In the upper half plane, this corresponds
to summing over CM-points (for negative discriminant), or integrating
along geodesics (for positive discriminant).
The link with zeta functions is provided by a result of Hecke: an
Eisenstein series is toroidal if its weight is a zero of the zeta
function of the corresponding field.
We consider this theory for function fields of certain curves over finite fields. The toroidal integrals corresponds to sums over certain vector bundles on the curve.
We compute the space of toroidal automorphic forms for the function field of three elliptic curves over finite fields. The method is elementary: we reduce the vanishing of toroidal integrals to an infinite system of linear equations on some graph. For this, one has to understand the moduli of certain vector bundles on the elliptic curve.
We deduce an "automorphic'' proof for the Riemann hypothesis for the zeta function of those curves. Joint work with Oliver Lorscheid (arxiv:math/0710.2994).
Title: An effective "Uniqueness Principle" in the theory of Eisenstein Series on higher rank groups
Title: Modular forms and elliptic curves over Q(zeta_5)
There have been many computational investigations into modularity of elliptic curves over number fields other than Q, especially for imaginary quadratic fields (Cremona and his students, Dieulefait--Guerberoff--Pacetti) and real quadratic fields (Dembele). In this talk we present work in progress with F. Hajir, D. Ramakrishnan, and D. Yasaki that treats the totally complex quartic field of fifth roots of unity. We will discuss our computational techniques and will give examples of elliptic curves over this field whose L-series apparently match those given by the Hecke data of eigenforms.