Date:
Thu, 03/11/202216:00-17:15
Location:
Ross 70
Link to recording: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=830f030c-524a-4e73-94a6-af3400938297
Link to recording previous for week: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=9694ea89-b9f3-4af8-a951-af34009381ff
Title -
Optimal strong approximation and the Sarnak-Xue density hypothesis (lecture 2)
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 e ≥ 3/2 . 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.
Link to recording previous for week: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=9694ea89-b9f3-4af8-a951-af34009381ff
Title -
Optimal strong approximation and the Sarnak-Xue density hypothesis (lecture 2)
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 e ≥ 3/2 . 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.