Project Details
Projekt
FOR 570: Algebraic Cycles and L-functions
Subject Area
Mathematics
Term
from 2005 to 2012
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 5471524
The research project belongs to the intersection of number theory and algebraic geometry inside mathematics. Basic for number theory is the problem to solve algebraic equations and to make as far as possible quantitative and qualitative statements about these solutions. For this purpose, it has become useful to encode the number of solutions (over finite fields for every prime number p) in the so-called L-function. Many deep statements can be or should be read of from the properties of the L-function. We mention the Riemann conjecture and the Birch-Swinnerton-Dyer conjecture, two of the seven millennium price problems of mathematics.
In 1989, Bloch and Kato conjectured, as far reaching generalisation of the Birch-Swinnerton-Dyer conjecture, a very precise connection between the special values of the L-function and certain algebra-geometric invariants of the underlying system of algebraic equations. This connection is very surprising due to the fact that the L-function is build out of only local information for all prime numbers p, but knows about global invariants, that do not only depend on p. So far, this important conjecture could be verified only in a very limited number of cases.
The goal of the Research Unit is on the one hand to verify the conjecture for further classes of algebraic equations and on the other hand to extend and build up further the theories, which enter the conjecture. This concerns motivic cohomology, etale cohomology and geometric class field theory, Iwasawa theory and Arakelov theory. Moreover, automorphic forms and polylogarithms play a decisive role. In addition, there are connections with topology, homotopy theory and secondary characteristic classes. The results obtained, are expected to have an impact on the theories involved.
In 1989, Bloch and Kato conjectured, as far reaching generalisation of the Birch-Swinnerton-Dyer conjecture, a very precise connection between the special values of the L-function and certain algebra-geometric invariants of the underlying system of algebraic equations. This connection is very surprising due to the fact that the L-function is build out of only local information for all prime numbers p, but knows about global invariants, that do not only depend on p. So far, this important conjecture could be verified only in a very limited number of cases.
The goal of the Research Unit is on the one hand to verify the conjecture for further classes of algebraic equations and on the other hand to extend and build up further the theories, which enter the conjecture. This concerns motivic cohomology, etale cohomology and geometric class field theory, Iwasawa theory and Arakelov theory. Moreover, automorphic forms and polylogarithms play a decisive role. In addition, there are connections with topology, homotopy theory and secondary characteristic classes. The results obtained, are expected to have an impact on the theories involved.
DFG Programme
Research Units
Projects
- A1-Homotopietheorie und Arithmetik (Applicant Schmidt, Alexander )
- Arithmetische Erweiterungen und Bierweiterungen für algebraische Zykel (Applicant Künnemann, Klaus )
- Arithmetische Erweiterungen und ihre Ext-Gruppen (Applicant Künnemann, Klaus )
- Endlichkeitssätze in der motivischen Kohomologie (Applicant Jannsen, Uwe )
- Homotopieinvarianten arithmetischer Schemata (Applicant Schmidt, Alexander )
- Iwasawa Theorie und p-adische L-Funktionen (Applicant Huber-Klawitter, Annette )
- Neue Kohomologietheorien in Charakteristik p und 0 (Applicant Jannsen, Uwe )
- Polylogarithmen und spezielle Werte von L-Funktionen (Applicant Kings, Guido )
- Tamagawazahlen, p-adische Regulatoren und spezielle Werte von L-Funktionen (Applicants Huber-Klawitter, Annette ; Kings, Guido )
- Triangulierte Motive und A^1-Homotopietheorie (Applicant Huber-Klawitter, Annette )
- Zentralprojekt (Applicants Huber-Klawitter, Annette ; Kings, Guido )
Spokesperson
Professor Dr. Guido Kings
