Geometric and Analytic Number Theory

• Applicants: Professor Dr. Valentin Blomer; Professor Dr. Jörg Brüdern; Professor Dr. Ulrich Derenthal
• Subject Area: Mathematics
• Term: from 2014 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 255083470

The study of integers and their properties is as old as mathematics, and continues to fascinate human beings all through the world. This includes in particular prime numbers and integral solutions of polynomial equations. In modern mathematics, these are also linked with many other research directions. Our project concerns three of the most lively and important areas of research in modern analytic number theory, which are deeply interrelated. An important common feature of all three is the presence of deep links with geometry. In some cases, algebraic geometry appears as an important tool, and in their as a source of natural questions and problems. Conversely, number theory may be a source of insights and solutions of important questions in geometry. The applicants are among the top experts in the world in each of the intended projects, and their collaboration has already shown to be very fruitful in their previous SNF-DFG project. The first topic is the theory of trace functions over finite fields and their applications. This is the best-known meeting ground of algebraic geometry and analytic number theory, where the power of the general form of the Riemann Hypothesis over finite fields, due to Deligne, can lead to spectacular concrete applications. In addition, the trace functions are themselves remarkable objects and well-worth studying in their own right. When taken as a whole, their statistical properties are very rich. The project will explore all these directions. The second topic concerns the very old problems surrounding the solutions of Diophantine equations, in the case when there are many solutions, so that it is natural to count these and to study how they are distributed. Wide-ranging conjectures concerning this problem were proposed by Manin and his collaborators in the 1980s, and these brought to the fore many relations with the geometry of algebraic varieties. Subsequent work has also revealed many new phenomena. Our project intends to explore new examples, as well as establishing new tools to study these questions. The third topic deals with modular forms, automorphic forms, their associated L-functions, and especially with their analytic properties. These functions lie at the heart of modern number theory since the late 1960s, when R. Langlands set up the general philosophy and framework now known as the Langlands Program. The analytic study of automorphic L-functions was begun in earnest in the 1990s. Current developments, and our own research plans, include powerful techniques in higher rank, as well as systematic exploration of the properties of families of L-functions. The natural interactions between the three projects lies not only in direct links, such as applications of trace functions and exponential sums to the circle method or to automorphic forms, but also on the fundamental similarity of scientific outlook, and of methods, that are shared by all members of the project and their respective groups.

• DFG Programme: Research Grants
• International Connection: Switzerland
• Co-Investigators: Professor Dr. Emmanuel Kowalski; Professor Dr. Philippe Michel