Integrality of Stickelberger elements
Final Report Abstract
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 investigated basic properties of the occurring analytic objects which are necessary to enable an action on the class group. These so-called Stickelberger elements have rational coefficients and so have to be multiplied by a suitable constant to obtain integral coefficients. Assuming the relevant special case of the equivariant Tamagawa number conjecture we determined what the best possible constant should be. Then we verified this in many cases (it suffices to assume that the underlying Galois group is nilpotent). In the same cases we proved that the Stickelberger elements times this constant indeed annihilate the class group. We also have analogous results for higher K-groups. Finally, we have related these elements to p-adic L-series - an entirely different kind of analytic object. This allowed us to verify a new interesting special case of a conjecture of Gross, which relates values of complex and p-adic Artin L-series at 0.
Publications
- Integrality of Stickelberger elements, Ph.D. thesis, Universität Duisburg-Essen, 2021
N. Ellerbrock
(See online at https://doi.org/10.17185/duepublico/74822) - Integrality of Stickelberger elements and annihilation of natural Galois modules
N. Ellerbrock and A. Nickel
(See online at https://doi.org/10.48550/arXiv.2203.12945)
