Basic notions Seminar - Shai Evra (HUJI) - Optimal strong approximation and the Sarnak-Xue density hypothesis

Thu, 27/10/202216:00-17:15
Ross 70
Link to recording: 
Title -

Optimal strong approximation and the Sarnak-Xue density hypothesis 

Abstract -

It is a classical result that the modulo map, S L 2 ( Z )S L 2 ( Z /q Z ) , is onto for any q N . The generalization of this phenomenon to other arithmetic groups goes under the name of strong approximation. The following natural question was raised in a letter of Sarnak: What is the minimal exponent e>0 , such that for any large q , almost any element of S L 2 ( Z /q Z ) has a lift in S L 2 ( Z ) with coefficients of size at most q e ? A simple pigeonhole principle shows that e3/2 . In his letter Sarnak proved that this is in fact tight, i.e. e=3/2 . Call this optimal strong approximation for S L 2 ( Z ) . The proof relies on a density theorem of the Ramanujan conjecture for S L 2 .

In our first talk we give a brief overview of the strong approximation, a quantitative strengthening of it called super strong approximation, and the above mentioned optimal strong approximation phenomena, for arithmetic groups. We highlight the special case of p -arithmetic subgroups of classical definite matrix groups and describe a connection between the optimal strong approximation property and the optimal almost diameter property for finite quotients of Bruhat-Tits buildings.

In our second talk we state the Sarnak-Xue density hypothesis and show how it implies the optimal strong approximation. We then give a (very) brief introduction to the theory of automorphic representations, the Langlands program, the endoscopic classification of Arthur and the generalized Ramanujan conjecture. Finally we show how special instances of the Sarnak-Xue density hypothesis can be deduced from recent advances in the Langlands program.