Ramanujan Conjecture and the Density Hypothesis - Shai Evra

Explore the intricacies of number theory in this seminar talk delivered by Shai Evra from Princeton University at the Joint IAS/Princeton University Number Theory Seminar. Delve into the evolution of the Ramanujan Conjecture, examining its various forms including the Generalized Ramanujan Conjecture (GRC) and the Generalized Ramanujan-Peterson Conjecture (GRPC). Investigate the Samak-Xue Density Hypothesis (SXDH) and its implications. Gain insights into the automorphic classification of general linear and classical groups, as well as the concept of automorphic genus. Examine key claims and examples, with a focus on a main example that illustrates the discussed concepts.

Intro
Arithmetic Spaces
Arithmetic Complexes
Purpose of this talk
Evolution of the Ramanujan Conjecture
Generalized Ramanujan Conjecture (GRC)
Generalized Ramanujan-Peterson Conjecture (GRPC)
Nahve Ramanujan Conjecture (NRC)
Samak-Xue Density Hypothes's (SXDH)
Samak-Xue Density Hypothesis (SXDH)
Remarks on (SXDH)
Automorphic Classification of General Linear Groups
Automorphic Classification of Classical Groups
Automorphic Genus
Claims and Eamples
Main Example