Periodic Reporting for period 1 - AuForA (Automorphic Forms and Arithmetic)
Reporting period: 2023-04-01 to 2025-09-30
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.
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.
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
- 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