September 11
Speaker: Irena Wang
Title: Drinfel'd Modules and a Function Field Analog to Atkin's Orthogonal Polynomials
Abstract: Elliptic curves and rank-two Drinfel'd modules are a well-known example of the numerous parallels between number theory
over number fields and over function fields. In particular, similarly to elliptic curves, the endomorphism ring of a Drinfel'd module is either isomorphic to a ring
F_q[T], an order in an imaginary extension of F_q(T), or an order in a quaternion extension of F_q(T), in which case it is called "supersingular."
It's well-known that, over F_q[T], there exists a finite number n_q of isomorphism classes of supersingular Drinfel'd modules (each identified with
a unique supersingular j-invariant), which motivates the construction of a polynomial product called the supersingular locus \prod{x-j), over all such
isomorphism classes. In this talk, I'll present a result giving an explicit construction of polynomials congruent to the supersingular loci for
arbitrary n_q, which was proved by Doris Dobi, Nick Wage, and I in 2007. This result is an adaptation of a similar method presented in a famous paper
of Kaneko and Zagier, which used continued fraction convergents to generate "Atkin Orthogonal Polynomials" congruent the the supersingular loci for
elliptic curves.
September 18
Speaker: Tonghai Yang (University of Wisconsin)
Title: Faltings heights of CM cycles and derivatives of L-functions.
Abstract: In this talk, I will explain a systematic way of constructing Green function for Kudla's special cycles on a Shimura variety. Then we give an
explicit formula for their CM value, which inspires two conjectures about Faltings height of CM cycles, central derivative of L-series, and arithmetic
intersections. Finally, if time permits, we will give evidence of these conjectures and use our explicit formula to give a new proof of the Gross-Zagier
formula.
September 25
Speaker: Xinyi Yuan (IAS)
Title: Derivatives of triple product L-functions.
Abstract: I will talk about the conjecture of Gross-Shoen which relates the central derivative of the Rankin-Selberg L-function of three
modular forms to the Beilinson-Bloch height pairing of the diagonal cycle on the cubic power of the corresponding modular curve.
October 16
Speaker: Ben Howard (Boston College)
Title: Intersection theory on Shimura surfaces.
Abstract: Kudla has proposed a general program to relate arithmetic intersection multiplicities of special cycles on Shimura varieties to
Fourier coefficients of an Eisenstein series. The lowest dimensional case, in which one intersects two codimension one cycles on the integral model of
a Shimura curve, has been completed by Kudla-Rapoport-Yang. We will prove results in a higher dimensional setting: on the integral model of a Shimura
surface we consider the intersection of an embedded Shimura curve with a family of codimension two cycles of complex multiplication points, and
relate the intersection numbers to Fourier coefficients of a Hilbert modular form of half-integral weight.
October 30
Speaker: Bisi Agboola (UCSB)
Title: p-adic conjectures of Birch and Swinnerton-Dyer type for elliptic
curves with complex multiplication
Abstract: In this talk, I shall discuss the study of special values of the
Katz two-variable p-adic L-function that lie outside the range of
interpolation, and I shall explain how these are related to certain
conjectures of Birch and Swinnerton-Dyer type.
November 20
Speaker: Jeehoon Park (McGill University)
Title: Iwasawa main conjecture for CM elliptic curves at supersingular primes
Abstract: We generalize the Pollack-Rubin proof of the Iwasawa main
conjecture for CM elliptic curves over Q at supersingular primes to
CM elliptic curves over an abelian extension of the imaginary
quadratic field given by CM. We will explain the precise set-up, how
to construct plus/minus algebraic p-adic L-functions and plus/minus
analytic p-adic L-functions and how they coincide. This is a joint
work with Byoung Du Kim and Bei Zhang.