Abstract. Let Y be a compact Riemann surface with a Riemannian metric
of constant curvature −1. Consider the corresponding Laplace-Beltramy
operator in the space of functions on $Y$ and fix its eigenfunction $\phi$. Such
function is called Maass form - this is a special case of an automorphic form.
The study of automorphic forms and finding bounds on invariants arising
from these forms plays central role in many problems in Geometry and
Analytic Number Theory.
I will present a general method how to bound some invariants arising from
Maass forms. This method is based on representation theory of the group
$SL(2, R)$.
In my talk I will concentrate on the following concrete problem. Fix a
closed geodesic $C\subset Y$ , denote by $f$ the restriction of a Maass form $\phi$ to $C$
and consider its Fourier series expansion $f=\sum_k a_k\textrm{exp}(kt\cdot 2\pi i)$. Coefficients $a_j$ in this expansion are given by period integrals
$\int_C f(t)\textrm{exp}(-kt\cdot 2\pi i)dt$ - this is a special case of automorphic peri-
ods. I will discuss the problem how to give good bounds for coefficients $a_k$
when $k$ tends to infinity.
One of the goals of my talk is to describe the relation between automorphic
periods and special values of L-functions; this relation provides highly non-
trivial information about the asymptotic behavior of automorphic periods.
In particular it suggests some initial bound of these periods (it is called the
convexity bound.)
The main goal of the talk is to show how to use representation theory to
prove some stronger version of convexity bound for automorphic periods. I
will also try to indicate how one can try to prove stronger ”subconvexity“
bounds for this type of problems.
My talk is essentially an introduction to a very fascinating area of Analytic
Number Theory. I will try to make it completely elementary by concentrat-
ing on very concrete problems. I do not assume any special knowledge of
automorphic theory.
This talk is an exposition of my joint works with Andre Reznikov.