Date:

Thu, 27/10/202216:00-17:15

Location:

Ross 70

__Link to recording: 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__

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.