# ANTHOS Sintesi della relazione

Project ID:
258713

Finanziato nell'ambito di:
FP7-IDEAS-ERC

Paese:
Germany

## Final Report Summary - ANTHOS (Analytic Number Theory: Higher Order Structures)

Modern analytic number theory studies arithmetic questions by means of highly structured and advanced tools. The project ANTHOS (Analytic Number Theory - Higher Order Structures) is an interdisciplinary approach to this area at the interface of number theory, algebraic geometry and automorphic forms. The key players in this project are classical mathematical objects:

• L-functions: certain analytic functions on the complex plane;

• automorphic forms: highly symmetric functions living on a Lie group;

• algebraic varieties: geometric objects like surfaces and higher dimensional analogues.

All of these encode in various ways deep arithmetic information that can be investigated by studying these objects. Conversely, arithmetic techniques can be used to study properties of L-functions, automorphic forms, and algebraic varieties. This interplay is one of the cornerstones of analytic number theory.

The most important achievements include progress towards the Ramanujan conjecture over number fields, a proof of Manin’s conjecture for a certain cubic fourfold along with several novel analytic counting techniques, the development of a convenient version of Kuznetsov’s formula for the group GL(3) along with various applications, the computation of moments of L-functions leading to new instances of subconvexity, and bounds for the sup-norm of automorphic forms on GL(n) for arbitrary n.

• L-functions: certain analytic functions on the complex plane;

• automorphic forms: highly symmetric functions living on a Lie group;

• algebraic varieties: geometric objects like surfaces and higher dimensional analogues.

All of these encode in various ways deep arithmetic information that can be investigated by studying these objects. Conversely, arithmetic techniques can be used to study properties of L-functions, automorphic forms, and algebraic varieties. This interplay is one of the cornerstones of analytic number theory.

The most important achievements include progress towards the Ramanujan conjecture over number fields, a proof of Manin’s conjecture for a certain cubic fourfold along with several novel analytic counting techniques, the development of a convenient version of Kuznetsov’s formula for the group GL(3) along with various applications, the computation of moments of L-functions leading to new instances of subconvexity, and bounds for the sup-norm of automorphic forms on GL(n) for arbitrary n.