Iwasawa theory of p-adic Lie extensions

• Applicant: Professor Dr. Otmar Venjakob
• Subject Area: Mathematics
• Term: from 2007 to 2011
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 48575220

One of the most challenging topics in modern number theory is the mysterious relation between special values of L-functions and Galois cohomology: they are the shadows in the two completely different worlds of complex and p-adic analysis of one and the same geometric object, viz the space of solutions for a given diophantine equation over the integral numbers, or more generally a motive M. The main idea of Iwasawa theory is to study manifestations of this principle such as the class number formula or the Birch and Swinnerton Dyer Conjecture simultaneously for whole p-adic families of such motives, which arise e.g. by considering towers of number fields or by (Hida) families of modular forms. The aim of this project is to supply further evidence forI. the existence of p-adic L-functions and for main conjectures in (non-commutative) Iwasawa theory,II. the (equivariant) e-conjecture of Fukaya and Kato as well asIII. the 2-variable main conjecture of Hida families.In particular, we hope to construct the first genuine non-commutative p-adic L-function as well as to find (non-commutative) examples fulfilling the expectation that the e-constants, which are determined by the functional equations of the corresponding L-functions, build p-adic families themselves. In the third item a systematic study of Lie groups over pro-p-rings and Big Galois representations is planned with applications to the arithmetic of Hida families.

• DFG Programme: Research Grants