In number theory, a period is a complex number, whose real and imaginary part are the value of an integral of rational functions with rational coefficients, over a rational domain. Periods arise in various contexts of mathematics and play a decisive role in the study of special values of automorphic and motivic L-functions, as documented by famous conjectures of Deligne and Bloch-Kato.
In turn, knowledge about the algebraic nature of these L-values is the linchpin for the construction of p-adic L-functions. Showing their existence is a fundamental number-theoretical problem and an indispensable ingredient in every conceptual generalization of the Iwasawa Main Conjecture to groups of higher rank.
This START-project has two major visions:
(1) To explore new landmarks in the theory of periods and their connection to special values of L-functions and
(2) to carry out new and explicit constructions of classes of p-adic automorphic L-functions.
Our main focus will in fact be on the three eminent open conjectures of Deligne, Bloch-Kato and Iwasawa- Greenberg: We will propose a challenging, but precise strategy that allows to prove “automorphic analogues” of Deligne's conjecture in a great variety of cases; we will provide an ambitious project of research, which aims to establish substantial progress towards the “adjoint” Bloch-Kato conjecture, generalizing work of Hida, Diamond-Flach-Guo and Dimitrov to all dimensions; and we will present a clear strategy of how to construct new p-adic L-functions, preparing the ground for a generalization of the recent proof of the Iwasawa-Greenberg Main Conjecture by Skinner-Urban and Kato to general linear groups of arbitrary rank.
Aside of these key-aspects, and as an additional benefit of our project, we will also propose an array of different problems for coherent cohomology of Shimura varieties, which allow to embody the theory of coherent cohomology in an entirely automorphic context.
By its clear nature, our ambitious project shall be carried out in intensive collaboration with a strong team of researchers. Our local team at the University of Vienna, guided and supported by the expertise and experience of the principal investigator (HG), consists of a PhD-student and a postdoctoral researcher; whereas our international network of collaborators, a name-dropping list of first-class experts in the field, just substantiates this START-project's location at the current edge of international research.