Special L-values and p-adic L-functions

All is Number. (Pythagoras) In our modern world numbers are omnipresent. In technology, politics, economics, science, religion, in space and time numbers show us deep ways from disorder to order, from chaos to structure. In this way numbers significantly coin the progress of our society, of our prosperity and security. An excellent example is data-security indispensable in our modern information-society. To guarantee effective protection of sensitive personal information, such as data about communication, bank accounts or medical records, is the central task of data-encryption. Two of todays most important encryption methods are Elliptic Curve Cryptography (ECC) and RSA-encryption. The first technique is used to protect sensitive items of daily life, such as debit cards, passports and health-insurance cards; the latter one ensures save internet- and telephone-connections, is used to encrypt emails and to protect online-banking. Both encryption methods are based on fundamental and fascinating results of number theory: ECC uses the complexity of rational points of infinite order on elliptic curves; while RSA is based on one of the deepest open problems of mathematics, the distribution of prime numbers. To understand and explore these two mathematical structures is a century-old, eminent problem of mathematics. In modern number theory we know that the description of these structures is encoded by highly complex functions, called L-functions. To understand these L-functions is the major goal of modern research in number theory. This START-project is devoted to tackling the major challenges of an in-depth investigation of these L- functions in a pioneering, ambitious and innovative way. The present START-project is based on ground- breaking recent results of the applicant and his collaborators and aims for a break-through on various frontiers of international, number-theoretical research. The mysterious behaviour of L-functions is part of several famous conjectures of number theory. This START-project has the ambitious goal to establish some of these conjectures in a conceptual and far- reaching context and to break new scientific ground in fundamental research.

"Number is the ruler of forms and ideas, and the cause of gods and daemons." (Pythagoras; according to Syrianus "On Aristotle Metaphysics", book 13) Number theory, the oldest discipline of mathematics, appreciates the enigmatic beauty of numbers and their internal relations. In this regard, a core-topic of number theory is to explore and describe the prime numbers - those mysterious numbers which can only be divided by 1 and themselves. The reason for which prime numbers still remain mysterious is that still it is not entirely clear how they are distributed among all numbers: Sometimes there are many prime numbers in short succession. Then, all in a sudden, they become very seldom and almost seem to disappear completely. However, we know since Euclid's days (300 b.c.) that there are infinitely many prime numbers, i.e., their progression never stops and so the above phenomena will reappear again and again. What is the reason for this mysterious behavior? In modern mathematics we have developed powerful techniques and approaches to deal with this - yet unsolved! - problem: For instance, we know now that certain functions, the so-called "L-functions", encode the mysterious distribution of prime numbers. And yet, these L-functions have withstood all attempts to subject them to a complete description. More accurately said, like prime numbers themselves, until today the exact properties of L-functions remain a book sealed with seven seals. In our START-project we shed significant new light on yet unknown properties of L-functions. As one of our major achievements, we could describe important classes of L-functions at so-called "values": These special values have been a main object of research for more than 300 years, when Euler (known for Euler's constant e, which is the basis for any exponential growth) managed to provide the world's first formula for such special L-values in their most basic case. Our new results allowed us to solve several old and also recent problems in number theory. For instance, together with our collaborators we could finally prove that a famous conjecture of P. Deligne from 1979 is true, if certain L-functions are non-zero at the the point s=1/2 - a property which is widely expected to hold in full generality. Our START-project should hence also be taken as a hint for why the study of L-functions is a fruitful and promising undertaking, always ready to set new cornerstones in modern mathematics. We are looking forward to reporting on many other new developments, which are based on this START-prize's results, in the future.