Automorphic Forms and Arithmetic

Project Information

AuForA

Grant agreement ID: 101054336

DOI

10.3030/101054336

EC signature date 5 September 2022

Start date 1 April 2023

End date 31 March 2028

Funded under

• European Research Council (ERC)

Total cost

€ 1 956 665,00

EU contribution

€ 1 956 665,00

Coordinated by

RHEINISCHE FRIEDRICH-WILHELMS-UNIVERSITAT BONN
Germany

Periodic Reporting for period 1 - AuForA (Automorphic Forms and Arithmetic)

Reporting period: 2023-04-01 to 2025-09-30

Summary of the context and overall objectives of the project

The golden thread of AuForA is the connection between fundamental arithmetic problems and the analytic theory of automorphic forms. The overarching goal is a cross-fertilization in both directions: genuine number theoretic problems will be solved through automorphic techniques, and automorphic forms and their L-functions will be investigated with number theoretic tools. Automorphic forms are interdisciplinary in nature, and parts of this proposal have natural links also to representation theory and harmonic analysis. Automorphic forms and arithmetic meet most intimately in the Langlands program: it is a fundamental conjecture that all "interesting'' arithmetic, i.e. Galois-theoretic, objects should be connected to analytic objects coming from automorphic forms, and there should be certain reciprocity laws relating them. One of the most powerful tools in the theory of automorphic forms are trace formulae which provide a sophisticated dictionary between spectral and geometric properties of locally symmetric spaces. The goal of AuForA is to explore and develop the rich theory at the interface of between number theory and automorphic forms for higher rank groups. This is motivated by several important conjectures such as the joint equidistribution conjectures of Michel and Venkatesh, the "beyond endoscopy'' program of Langlands and the density conjectures of Sarnak.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Documented by numerous high level journal articles and preprints, extensive research at the interface of number theory and automorphic forms has been performed with a focus on cross-fertilization in both directions. This includes in particular progress on several important conjectures in the field such as the joint equidistribution conjectures of Michel and Venkatesh, the "beyond endoscopy'' program of Langlands and the density conjectures of Sarnak. In particular, new density theorems have been establish along with a variety of arithmetic and analytic applications. This involves an intensive study of (relative) trace formulae and Kloosterman sums which is of independent interest and led to a number of independent applications. Several equidistribution problems have been solved successfully, a new key tool being a sophisticated lattice point counting technique. An imoportant hinges between automorphic forms and their number theoretic information are L-functions, far-reaching generalizations of the famous Riemann zeta-function. Several new results on L-functions have been proved.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

A representative sample of new results include:
- new density theorems for GL(n) with arithmetic applications to optimal lifting and analytic applications to zeros of L-functions
- new spectral summation formulae for GSp(4) with applications
- a proof of the mixing conjecture of Michel and Venkatesh under GRH with an effective rate of convergence and an unconditional proof of the unipotent mixing conjecture
- sharp lattice point counts in higher rank
- strong bounds for general Kloosterman sums on GL(n), as well as distributional properties of classical Kloosterman sums
- new results on higher rank L-functions
- a novel approach to classical shifted convolution problems
- far-reaching progress in the sup-norm problem in unbounded rank based on a powerful interplay between number theory and harmonic analysis