- Vivi Padova
- Il Bo
ARGOMENTI: International Area
Announce of the Advanced Course on Arithmetic Geometry for
Function Fields of Positive Characteristic, which will take place from February 22 to March 5, 2010 at the Centre de Recerca Matemàtica, Bellaterra.
The main focuses of the advanced courses are Iwasawa theory and Characteristic p L-functions. Iwasawa theory techniques, which had among their sources the arithmetic of constant function field extensions, were later brought back to the characteristic p world: for example, Crew proved a geometric version of Iwasawa's main conjectures already twenty years ago, which has been recently
generalized to the non-commutative setting by Burns.
Iwasawa theory of abelian varieties is strictly linked to the Birch and Swinnerton-Dyer conjecture: among many recent advances in this direction, we mention Ulmer's results on Gross-Zagier formulas and on the rank conjecture.
As for L-functions, some of the most striking developments in this field are the following: Goss' invention of "characteristic p-valued" L-functions, Anderson's generalization of Drinfeld modules to t-motives, the many spectacular results on transcendence of zeta-values, Thakur's gamma functions, Taguchi and Wan's work on meromorphic continuation of Goss' L-functions and its extension by Böckle. Another important result is Böckle's construction of an Eichler-Shimura isomorphism for Drinfeld modular forms: together with his previous work with Pink on "crystals", this allows to move from cusp forms to L-series.
The advanced courses give a comprehensive description of a big part of the above topics, with emphasis in: Geometric Iwasawa main conjecture, arithmetic of Jacobians over function fields (BSD, rank,...), the arithmetic of Gamma, Zeta, multizeta values in function fields, and a cohomology theory of crystal for functions fields.
Important deadlines are December 22 (grant applications) and February 5 (registration and payment).