Programm on Arithmetic Algebraic Geometry

ESI, Vienna, January 2 - February 18, 2006

Organizers: Stephen S. Kudla, Michael Rapoport, Joachim Schwermer

This program will focus on several aspects of arithmetical algebraic geometry with an emphasis on the relation among algebraic cycles on Shimura varieties, special values of L-functions and automorphic forms.

I. Applications of recent developments in Rankin-Selberg theory to Arithmetic

The classical Rankin-Selberg method, which expresses the L-function attached to a pair of modular forms for as an integral against an Eisenstein series has played a very prominent role in the study of the arithmetic of special values of L-functions. In fact, the famous conjectures of Deligne concerning the special values of L-functions of motives at their critical points were originally made in response to examples obtained by Zagier using such Rankin-Selberg integrals. Both, Hecke's original "Mellin transform" integrals and the Rankin-Selberg integrals were carried over into the theory of automorphic representations by Jacquet and Langlands. In recent years and in the language of automorphic representations, this classical method has been vastly extended by the work of Rallis, Piatetski-Shapiro, Gelbart, Bump, Ginzburg, Soudry, Jiang and others to provide integral representations, and hence, analytic information about many new families of automorphic L-functions. While in the analytic theory of these new Rankin-Selberg integrals has become quite well developed, relatively little work has been done on their arithmetic applications. A main goal of this part of the program is to stimulate research on such applications by bringing together specialists in the Rankin-Selberg methods and specialists in releveant areas of arithmetic and geometry.

II. Algebraic cycles on Shimura varieties

Shimura varieties, the varieties whose complex points arise as (unions of) quotients of bounded symmetric domains by arithmetic groups, are among the most interesting of all algebraic varieties, and are deeply tied to the theory of automorphic forms. Many, including the classical modular and Shimura curves and the Siegel modular varieties, are moduli spaces for families of abelian varieties. The structure and arithmetic of such varieties has been the subject of very extensive research. The main focus of this part of the program will be on the study of algebraic cycles on Shimura varieties, particularly on those "special cycles" which arise as sub-Shimura varieties. For example, one would like to calculate the heights of such cycles with respect to natural metrized line bundles. One would like to understand the classes of such cycles in (i) cohomology (topological) (ii) in the Chow ring (algebraic) (iii) in the arithmetic Chow ring (arithmetic) of a Shimura variety as well as the relation of such classes to special values of certain L-functions. Although the systematic study of such cycles is still in its early stages, there have been a number of recent developments (generalized Arakelov theory, modular generating functions, Lefschetz type restriction theorems) which suggest that important new progress is now possible. A goal of this part of the program is to stimualte collaborative research by gathering specialists currently working in this area.

Participants

Freydoon Shahidi (Purdue) Jan-Hendrik Bruinier (Koeln)
Guenter Harder (Bonn) Jan Nekovar (Paris)
Don Blasius (UCLA) Juerg Kramer (Berlin)
Erez Lapid (Hebrew U.) Stefan Mueller-Stach (Mainz)
Pierre Colmez (Paris) David Soudry (Israel)
Eric Urban (Columbia) Christopher Skinner (University of Michigan)
Wenzhi Luo (Ohio State) Masao Tsuzuki (Sophia University)
Dihua Jiang (Minnesota) Benjamin Howard (Boston College)
Jim Cogdell (Ohio State) Takahiro Hayata (Yamagata University)
Michael Harris (Paris) Wee Teck Gan
Christophe Cornut (Paris) Jean-Benoit Bost (Orsay)
Jean-Pierre Wintenberger (Strasbourg) Massimo Bertolini
Andreas Langer (Exeter) Tobias Berger (MPI Bonn)
Eric Urban (Columbia) Klaus Kuennemann (Regensburg)
Jose Burgos (Barcelona) Ulf Kuehn (Humboldt Berlin)

Monday, Jan 23

9:30--10:30 Pierre Colmez, About the p-adic local Langlands correspondance for GL2(Q_p)
11:00--12:00 Dihua Jiang, Cusp Forms of Odd Special Orthogonal Groups, I (joint series with David Soudry)
15:00--16:00 Massimo Bertolini, Heegner points, Stark-Heegner points and special values of complex L-series, I
16:30--17:30 Don Blasius, A conjecture on images of automorphic Galois representations.

Tuesday, Jan 24

9:00--10:00 Ulf Kuehn, Arithmetic Geometry associated with Hecke's Eisensteinseries of weight 2
10:15--11:15 David Soudry, Cusp Forms of Odd Special Orthogonal Groups, II (joint series with Dihua Jiang)
11:45--12:45 Jean-Pierre Wintenberger, On Serre`s modularity conjecture
15:00--16:00 Jan Nekovar, The Euler system of CM points on Shimura curves
16:30--17:30 Masao Tsuzuki, Fourier coefficient of the Poincare dual form of a modular divisor on U(n,1)

Wednesday, Jan 25

9:30--10:30 Massimo Bertolini, Heegner points, Stark-Heegner points and special values of complex L-series, II
11:00--12:00 Chris Skinner, Arithmetic aspects of Eisenstein series, I
15:00--16:00 Jim Cogdell, ``Stability of gamma" as a useful tool
16:30--17:30 Erez Lapid, Periods of automorphic forms, III (periods over unitary groups)

Thursday, Jan 26

9:30--10:30 Chris Skinner, Arithmetic aspects of Eisenstein series, II
11:00--12:00 David Soudry, Stability of local gamma factors, arising from the doubling method
15:00--16:00 Christophe Cornut, Distribution of conjugated CM points in Product of Shimura curves
16:30--17:30 Wenzhi Luo, Equidistribution Problems in Homogeneous Varieties

Friday, January 27

9:30--10:30 Freydoon Shahidi, Langlands Functoriality Conjecture and Special Values of L-functions
11:00--12:00 Jean-Benoit Bost,
15:00--16:00 Klaus Kuenemmann,
16:30--17:30 Eric Urban, Hida Theory for GU(2,2) and the Eisenstein ideal