Automorphic models and L-values for global algebras

For a number theorist, prime numbers play the role, which atoms play for a physicist. Indeed, like all physical objects are built up by atoms, the prime numbers are the building-blocks of all natural numbers (= the numbers 1, 2, 3, 4, 5, ): Each such number can be written in a unique way as a product of prime numbers. However, even after 2500 years of research starting with Euclid the distribution of the prime numbers among all natural numbers is still a mystery and seems to follow no clear pattern (at least) at first glance. In todays number theory we know that certain special functions, called L-functions, provide the key to describe such a pattern, and indeed encode all what can be said about the distribution of prime numbers. The problem is that these L-functions seem equally mysterious like the prime numbers themselves and do not reveal their properties easily. The research project Automorphic models and L-values for global algebras aims to start a new wave of research on these (seemingly) unsearchable L-functions. The PI together with his team of young researchers (it is planned to hire two PhD-students from the funding of this project) proposes to extend and to generalize many important result of the recent past on this subject, hence substantially enlarging the map of the theory. The main focus of our work is on exploring the connections of the values of such L-functions at special points to so-called period-invariants, which are defined by a comparison of rational structures on certain model spaces. This will include a thorough and very broad analysis of these period-invariants and the aforementioned model spaces as well. The PI expects that the benefits for the continuously evolving theory of L-functions will be twofold: Firstly, our projected results will allow a much more conceptual analysis of certain special values of L-functions, therefor also shading completely new light on already existing, deep results. Secondly, by the novelty of our approach (we are the first in this context, who suggest to consider invertible elements of completely general, central simple algebras), we will lay a strong fundament for further groundbreaking research in number theory.

"La libertad es como un nmero primo." - Roberto Bolaño in Los Detectives Salvajes. Prime numbers are an integral part of today's life - although perhaps invisible to the layperson. Modern encryption methods such as the RSA-method or Elliptic Curve Cryptography protect our sensitive data (bank accounts, credit cards, passports, insurance cards) and protect us from misuse in online purchases and identity-theft. However, it is sheerly almost impossible to tell, whether a given number is prime or not. And it is in fact one of the biggest unsolved problems in mathematics, and particularly of its special discipline, number theory, to understand the distribution of prime numbers in the "infinite sea" of all numbers. Today, modern research knows that certain number-theoretically relevant functions, so-called L-functions, encode the distribution of prime numbers - albeit in a way that is still very mysterious to us. The aim of this research project was to describe the behavior of these L-functions in selected, important special cases and thus to gain essential new insights into arithmetic.