This is a joint work with {\bf Ronny Hadani}.
Suppose we want to solve an algebraic problem over the finite field
$F_q$, formulated in terms of a function $F$ defined on certain
set $Y$. Grothedieck’s {\it Sheaf to Functions Correspondence}
suggests that there exists a geometric object (an
$\ell$-adic sheaf) $\SF$ from which the function $F$ can be derived:
$$ \SF \leadsto F $$
The sheaf $\SF$ is defined over the algebraic closure $\FFp$,
and can be approached by standard cohomological techniques.
In the lecture I will explain how this principle is worked out inorder to solve the Rudnick-Kurlberg Conjecture in the theory ofquantum chaos.
In the above case the sheaf $\SF$ is the trace of the {\it Weil representation sheaf} $\SK$. The construction of that
sheaf carried out in an unpublished letter from Deligne to Kazhdan in 1982.