## Final Activity Report Summary - SUBCONVEXITY (New bounds for automorphic 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 translations 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 objective of the project was to make progress towards this weaker hypothesis, precisely to exhibit new bounds for automorphic L-functions.

Several new bounds have been found, and a new technique has been developed which has the potential of treating cases where previous approaches were unsuccessful. Among the consequences are a better understanding of solutions of quadratic equations in three integral variables and with integral coefficients. Geometrically speaking, this corresponds to a better understanding of lattice points on ellipsoids in three-dimensional space.

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 objective of the project was to make progress towards this weaker hypothesis, precisely to exhibit new bounds for automorphic L-functions.

Several new bounds have been found, and a new technique has been developed which has the potential of treating cases where previous approaches were unsuccessful. Among the consequences are a better understanding of solutions of quadratic equations in three integral variables and with integral coefficients. Geometrically speaking, this corresponds to a better understanding of lattice points on ellipsoids in three-dimensional space.