Advanced Course on

Arithmetic Geometry for Function Fields

of Positive Characteristic

An activity of an i-MATH Intensive Research Programme

Group picture NEW!
Lecture Notes 1 (Prof. Dinesh S. Thakur)
Lecture Notes 2 (Prof. Böckle)
List of participants with lodging arranged through the CRM
Programme
List of participants
Dates:	February 22 to March 5, 2010
Place:	Centre de Recerca Matemàtica (CRM), Bellaterra, Barcelona, Spain	How to reach the CRM

Goals

Advanced course 1: Arithmetic of Gamma, Zeta, Multizeta values in Function Fields,

Dinesh S. Thakur (University of Arizona). Eight lectures of one and half hour each.

The aim of this series of lectures is to give a comprehensive description of some work of the author and others on the topic of the title. We will explore these functions, their properties, functional equations, interpolations, their special values, their connections with Drinfeld modules and Anderson's t-motives, their periods, algebraic relations they satisfy and proofs by various methods that there are no more relations between them. We will also describe open problems in the area.

More precisely, the course will describe some of the results obtained in several articles on the subject available on author's web-page math.arizona.edu/~thakur (see also his book `Function Field Arithmetic' or David Goss book `Basic structures of Function Field arithmetic, for exposition), together with papers (1) Anderson, Brownawell, Papanikolas (Annals 2004),

(2) some papers by C. Y. Chang, M. Papanikolas, J. Yu and Author, available on archives and also (3) works by Denis, Mendes-France-Yao, Allouche, Thiery, Dammame- Hellagouarch etc.

Advanced course 2: Curves and Jacobians over Function Fields

Speaker: Douglas Ulmer (Georgia Tech.). Eight lectures of one and half hour each, or six lectures of two hours each.

The aim of this course is to discuss the arithmetic of curves and their Jacobians over global function fields of characteristic p. The topics will include:

The conjecture of Birch and Swinnerton-Dyer for Jacobians and its connection with the Tate conjecture for surfaces over finite fields.

Instances in which the conjectures can be proven.

Constructions of Jacobians with high analytic and algebraic ranks.

Explicit points in high rank situations.

Advanced course 3: A cohomology for function fields arithmetic, and applications.

Speaker: Gebhard Böckle (Essen-Duisburg Universität). Eight lectures of one and half hour each.

The aim of the course is to present a cohomological theory for schemes in positive characteristic developed jointly with Richard Pink, and to give some applications of this theory. The main reference for this is [3].

A first application, and also the motivation for the construction of the above theory, is a purely algebraic proof of a conjecture of David Goss on the rationality of certain L-functions attached to families of t-motives. This result had been obtained first around 1995 by Taguchi and Wan using analytic methods. 

A second application is to Goss' analytic L-functions. Here again the first decisive results are due to Taguchi and Wan. I shall explain how the cohomological formalism has implications on the special values at negative integers. Surprisingly the cohomological approach is also useful in the algorithmic computation of such values.

If time permits, I shall also talk on applications to Drinfeld modular forms and their associated Galois representations.  

References

[1] G. Böckle, Global L-functions over function fields. Math. Ann. 323 (2002), no. 4, 737-795.

[2] G. Böckle, An Eichler-Shimura isomorphism over function fields between Drinfeld modular forms and cohomology classes of crystals.

[3] G. Böckle, R. Pink, Cohomological Theory of Crystals over Function Fields

[4] V. Lafforgue, Valeurs spéciales des fonctions L en caractéristique $p$. (French) [Special values of $L$-functions in characteristic $p$] J. Number Theory 129 (2009), no. 10, 2600--2634.

Advanced course 4: On main conjectures in geometric Iwasawa theory and related conjectures.

Speaker: David Burns (King's College of London). Eight lectures of one and half hour each, or six lectures of two hours each.

(Goals of advanced course 4 coming soon).

Scientific Committee

Francesc Bars (Universitat Autònoma de Barcelona)

Gebhard Böckle (Universität Duisburg-Essen)

David Burns (King's College London)

David Goss (Ohio State University at Columbus)

Ignazio Longhi ((From January 2010 CRM)

Fabien Trihan (University of Nottingham)

Xavier Xarles (Universitat Autònoma de Barcelona)

Douglas Ulmer (Georgia Tech)

Speakers

Registration

Registration fee: 300€

Registration includes: attendance to the lectures, documentation package, copy of the course notes (in case of advanced courses), social dinner, cultural activity, and coffee breaks

Available Grants

CRM Grant

Young researchers may apply for a grant and take advantge of a reduced registration fee and a lodging grant.
Awards are determined based on academic criteria and/or country of residence (giving special attention to advanced researchers from less favored countries).
There are two grants available:

• Full Grant: access to reduced registration + lodging (during the days of the activity in a shared apartment)
• Registration Grant: access to reduced registration

Reduced registration fee for either type of grant holders: 90€

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

• Registration and application for financial support

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.