Title:
The Sarnak–Xue Density Hypothesis for SO_5


Abstract:

A Naive version of the Ramanujan Conjecture (NRC) states that any cuspidal automorphic representation is tempered locally everywhere. While this is still believed to be true for GL_n, there are known counterexamples for the NRC for classical groups. The Sarnak–Xue Density Hypothesis (SXDH) states (in a precise way) that the number of violations to the NRC is small. It is expected that in applications the SXDH should act as a replacement for the NRC (similar to the role that the Bombieri-Vinogradov Theorem serves for the Riemann Hypothesis). 

The SXDH was proved by Sarnak—Xue for anisotropic  inner forms of SL_2 in the 90’s, and recently Assing—Blomer and Jana—Kamber proved a variant of the SXDH was proved for SL_n for any n. However, for other groups (especially anisotropic groups) the SXDH is still wide open. In this talk we will define the SXDH and a special case of it, called the Cohomological SXDH (CSXDH), and state our main result, the proof of the CSXDH for certain anisotropic inner forms of SO_5, the split group of 5 by 5 special orthogonal matrices. We also give some applications of the CSXDH, such as bounding the first Betti number of congruence hyperbolic 4-manifolds.

The main tool we use in our proof is Arthur’s endoscopic classification of automorphic representations of classical groups, combined with other results coming from the Langlands program, such as the Generalized Ramanujan-Petersson Conjecture for GL_n.  We give a generalization of the SXDH using Arthur’s language of A-parameters, which implies the classical SXDH, and prove it for our inner forms of SO_5.

This is based on a joint work Mathilde Gerbelli-Gauthier and Henrik Gustafsson.


Zoom Link: https://huji.zoom.us/j/87225685091?pwd=tDVUW6g3nHV42btR3ZnudsD3bc8PdT.1