Abstract:
Automorphic functions and representations are central for Number Theory. Automorphic Frobenius functional amounts to evaluation of automorphic functions the identity. From a point of view of Functional Analysis it is natural to determine a norm of Automorphic Frobenius functional as a functional on an (infinite dimensional) irreducible unitary automorphic representation. Such bounds provide bounds on values of automorphic functions, and sometimes could be translated into bounds on important number-theoretic quantities (e.g., on L-functions). I discuss recent approach of P. Nelson and A. Venkatesh (Acta Math. 226 (2021), no. 1, 1-209) which is based on a "geometrization" of this question via their quantitative version of the Orbit Method of A. Kirillov. This led them to a discovery of a "hidden symmetry" and an unexpected employment of Ratner's theorem. For simplicity, my discussion will be restricted to the group SL(2,R) while the general case is discussed in arXiv:2408.10793.