Integrality of Stickelberger elements

• Applicant: Professor Dr. Andreas Nickel
• Subject Area: Mathematics
• Term: from 2018 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 399131371

The class group of a number field is one of its most interesting arithmetic invariants. The analysis of its structure is a central theme of research in number theory for a long time. It is conjectured that certain analytic objects act on and indeed annihilate the class group, giving some constraints on its structure. In this project we like to investigate basic properties of the occurring analytic objects which are necessary to enable an action on the class group. These so-called Stickelberger elements should then be compared to p-adic L-series - an entirely different kind of analytic object. From this we like to deduce in as many cases as possible that the Stickelberger elements indeed annihilate the class group. If possible we even like to prove new cases of the considerably stronger "equivariant Tamagawa number conjecture" which also predicts a close relation between certain analytic and arithmetic objects.For this we have to study the functorial properties of Stickelberger elements and of the occurring algebraic structures. In particular, the behaviour in infinite towers of number fields will play a pivotal role in order to apply methods of Iwasawa theory. This should put us in a position to deduce new cases of the aforementioned conjectures. However, Stickelberger elements are only interesting for totally complex number fields. Examining appropriate conjectures for not necessarily totally complex number fields might then be a further topic of research.

• DFG Programme: Research Grants