Final Report Summary - AUTOMORPHIC (Automorphic forms and L-functions)
Automorphic forms are symmetric waves, generalisations of the well-known sine and cosine functions. The sine and cosine functions are defined on the line of real numbers, and their periodicity can be described as invariance under certain motions of the line. Automorphic forms are defined on spaces with rich geometry, where more complicated symmetries are present. They are the harmonic components of symmetric functions on the space, similarly as nice periodic functions on the line can be decomposed into sines and cosines. Mysteriously and magically, automorphic forms can make very deep properties of the integers visible. L-functions are useful in formulating, conjecturing and in some cases proving such properties. In short, automorphic L-functions provide a certain key to understanding the integers.
A famous unproved property of automorphic L-functions concerns the distribution of their zeros. Namely, it is expected that all nontrivial zeros of L-functions are located on a certain line of the plane of complex numbers, the axis of symmetry of the L-function. This property is the Riemann hypothesis, one of the most important unsolved problems in mathematics. It has a number of deep and interesting consequences, a notable one being the Lindelöf hypothesis stating that automorphic L-functions are not too large on their axis of symmetry. The main objective of the project was to make progress towards this weaker hypothesis, precisely to exhibit new bounds for automorphic L-functions. Closely related secondary objectives were a better understanding of the average size of automorphic L-functions and improved bounds for the underlying automorphic forms.
The project introduced several new ideas in the field, leading to strong and useful results in all three aspects mentioned above. First, a Burgess-like subconvex bound was proved for twisted Hilbert modular L-functions over totally real number fields. This result has an application for Hilbert's eleventh problem: the number of solutions of quadratic equations in three algebraic integers can be estimated more precisely than before. Second, a hybrid asymptotic formula was established for the second moment of Rankin-Selberg L-functions. A surprising aspect of the asymptotic formula is the appearance of a secondary main term, which seems to be a new phenomenon of the underlying family of L-functions. Third, it was shown that Hecke-Maass wave forms of square-free level do not have large peaks, improving significantly on previous results in the subject.
The main results of the project were achieved in collaboration with leading experts Valentin Blomer (Göttingen) and Nicolas Templier (Princeton). Additional leading experts James Cogdell (Columbus, Ohio) and Guillaume Ricotta (Bordeaux) were invited for short periods of consultation and joint research.
A famous unproved property of automorphic L-functions concerns the distribution of their zeros. Namely, it is expected that all nontrivial zeros of L-functions are located on a certain line of the plane of complex numbers, the axis of symmetry of the L-function. This property is the Riemann hypothesis, one of the most important unsolved problems in mathematics. It has a number of deep and interesting consequences, a notable one being the Lindelöf hypothesis stating that automorphic L-functions are not too large on their axis of symmetry. The main objective of the project was to make progress towards this weaker hypothesis, precisely to exhibit new bounds for automorphic L-functions. Closely related secondary objectives were a better understanding of the average size of automorphic L-functions and improved bounds for the underlying automorphic forms.
The project introduced several new ideas in the field, leading to strong and useful results in all three aspects mentioned above. First, a Burgess-like subconvex bound was proved for twisted Hilbert modular L-functions over totally real number fields. This result has an application for Hilbert's eleventh problem: the number of solutions of quadratic equations in three algebraic integers can be estimated more precisely than before. Second, a hybrid asymptotic formula was established for the second moment of Rankin-Selberg L-functions. A surprising aspect of the asymptotic formula is the appearance of a secondary main term, which seems to be a new phenomenon of the underlying family of L-functions. Third, it was shown that Hecke-Maass wave forms of square-free level do not have large peaks, improving significantly on previous results in the subject.
The main results of the project were achieved in collaboration with leading experts Valentin Blomer (Göttingen) and Nicolas Templier (Princeton). Additional leading experts James Cogdell (Columbus, Ohio) and Guillaume Ricotta (Bordeaux) were invited for short periods of consultation and joint research.