Advanced Course on

Shimura Varieties and L-functions

An activity of an i-MATH Intensive Research Programme

Group Picture (new!)
Prof. Zhang's Notes
Programme
List of participants with lodging arranged through the CRM
List of Participants
Dates:	October 19 to 24, 2009
Place:	Centre de Recerca Matemàtica (CRM), Bellaterra, Barcelona, Spain	How to reach the CRM

Goals

Arithmetic of Shimura curves and the Birch and Swinnerton-Dyer conjecture, by Shou-Wu Zhang (Columbia University). Six lectures of two hours.

The aim of this series of lectures is to give a comprehensive description of some recent work of the author and his students on generalisations of the Gross-Zagier formula, Euler systems on Shimura curves and rational points on elliptic curves. More precisely, the course will describe some of the results obtained in the following articles:

1. X. Yuan, S. Zhang, W. Zhang, Heights of CM-points I: Gross--Zagier formula (http://www.math.columbia.edu/~szhang/papers/HCMI.pdf). This article provides a Gross-Zagier formula in a very general setting.

2. X. Yuan, S. Zhang, W. Zhang, Heights of CM-points II: Chowla--Selberg formula (In preparation). This note provides formulae for logarithmic derivatives of Dedekind zeta functions of totally real fields and CM-fields.

3. Y. Tian, S. Zhang, Euler systems of CM-points on Shimura curves (In preparation). This article gives a generalization of Kolyvagin's work and some applications to Diophantine equations.

4. X. Yuan, S. Zhang, W. Zhang, Triple product L-series and Gross--Schoen cycles (In preparation). This paper contains a formula for the derivative of the triple product L-series and a new construction of rational points on elliptic curves.

In order to describe the proof of the results in the above papers, during the course we shall introduce the following preliminary topics:
 

A. Canonical and integral models of Shimura curves (work of Drinfeld and Carayol).

B.Heights and Arakelov theory of arithmetic surfaces (work of Weil, Neron, Tate, Arakelov, Faltings, Deligne, Gillet--Soule).

C. Weil representations and generating series (work of Weil, Jacquet--Langlands, Shimuzu, Waldspurger, and Kudla).

D. Euler systems (work of Kolyvagin).

An introduction to the above circle of ideas may be found in the survey article

5. S. Zhang, Elliptic curves, L-functions and CM-points, Current developments

in mathematics, 2001, 179--219, Int.Press, Somerville, MA, 2002, which is available in the homepage of the author.

A conjecture of André and Oort, by Bas Edixhoven (Leiden University) and Andrei Yafaev (University College of London). Six lectures of two hours.

The aim of the course is to give an introduction to the proof (under the generalised Riemann hypothesis) of the so-called Andre-Oort conjecture by Yafaev, Klingler and Ullmo.

More precisely, the main goal of the lectures will be to describe the results obtained by B. Klingler, E. Ullmo and A. Yafaev in the recent preprints
 

1. E. Ullmo, A. Yafaev, Galois orbits and equidistribution : towards the André-Oort conjecture, available at http://www.math.u-psud.fr/~ullmo/Prebublications/UllmoYafaev2.pdf
 

2. B. Klingler, A. Yafaev, The André-Oort conjecture, available at http://people.math.jussieu.fr/~klingler/papers.html

This conjecture says that if S is a Shimura variety and Z is any subset of special points of S, then the irreducible components of the Zariski closure of Z are sub-Shimura varieties. Important examples are the moduli spaces of polarised abelian varieties, where the special points are the points corresponding to abelian varieties with (sufficiently many) complex multiplications.

The course will follow the history of the subject, starting with the simplest non-trivial case, and keeping the most technical parts for the end. The main ingredients, Galois orbits, Hecke correspondences and equidistribution, will be introduced. A detailed sketch of the proof mentioned above will be given.

An introduction to the above topics can be found in the following references:

3. B. Edixhoven, A. Yafaev, Subvarieties of Shimura varieties. Ann. of Math. (2) 157 (2003), no. 2, 621--645.

4. R. Noot, Correspondances de Hecke, action de Galois et la conjecture d'André-Oort (d'après Edixhoven et Yafaev). Séminaire Bourbaki. Vol. 2004/2005. Astérisque No. 307 (2006), Exp. No. 942, vii, 165--197.
5. R. Pink, A combination of the conjectures of Mordell-Lang and André-Oort. Geometric methods in algebra and number theory, 251--282, Progr. Math., 235, Birkhäuser Boston, Boston, MA, 2005.

6. A. Yafaev, On a result of Moonen on the moduli space of principally polarised abelian varieties. Compos. Math. 141 (2005), no. 5, 1103--1108

7. A. Yafaev, A conjecture of Yves André's. Duke Math. J. 132 (2006), no. 3, 393--407.

Scientific Committee

Henri Darmon (McGill University, Montreal)

Fred Diamond (King's College of London)

Luis Dieulefait (Universitat de Barcelona)

Bas Edixhoven (Leiden University)

Victor Rotger (Universitat Politècnica de Catalunya)

Speakers

Shou-Wu Zhang, Columbia University

Arithmetic of Shimura curves and the Birch and Swinnerton-Dyer Conjecture
 

Andrei Yafaev, University College London

A Conjecture of André and Oort

Bas Edixhoven, Universiteit Leiden

A Conjecture of André and Oort

Registration and Financial Support

The CRM offers a limited number of grants covering registration and accommodation addressed to young researchers. Applications can be submitted during the registration process. The on-line registration system enables the following actions:

• Register to the activity.
• Apply for financial support. (Applicants will be asked, among other things, to upload a curriculum vitae. Please note that no applications will be taken into consideration otherwise.)
• Pay the registration fee.

You will be informed as soon as possible whether support is available.

Deadline for grant applications:	August 24, 2009
Deadline for registration and payment:	October 5, 2009

Registration fee: 300 Euros, including participation to the lectures, documentation package, a copy of the course notes, lunch tickets, a social dinner, a cultural activity, and coffee breaks.

For registration and payment fill out the documents above and follow the instructions there

Accommodation

Participants awarded with accommodation grants will have their lodging arranged through the organisation. The remaining participants are encouraged to book their lodging as soon as possible.

• Information on accommodation options for participants

For further information, please contact the CRM Administration.