The Generalized Ramanujan Conjecture (GRC) for GLn is a central open problem in modern number theory. Its resolution is known to yield several important applications. For instance, the Ramanujan-Petersson conjecture for GL2, proven by Deligne, was a key ingredient in the work of Lubotzky-Phillips-Sarnak on Ramanujan graphs.
One can also state analogues of (Naive) Ramanujan Conjectures (NRC) for other reductive groups. However, in the 70s Kurokawa and Howe-Piatetski-Shapiro proved that the (NRC) fails even for quasi-split classical groups.
In the 90s Sarnak-Xue put forth a Density Hypothesis version of the (NRC), which serves as a replacement of the (NRC) in applications.
In this talk I will describe a possible approach to proving the Density Hypothesis for definite classical groups, by invoking deep and recent results coming from the Langlands programme: The endoscopic classification of automorphic representations of classical groups due to Arthur, and the proof of the Generalized Ramanujan-Petersson Conjecture.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
