Abstracts for Number theory lunch seminar

In this talk, we consider local Maass forms.  These are functions which locally
satisfy the differential equation of a Maass form away from certain geodesics
of measure zero.  In joint work with Bringmann and Kohnen, a first natural
subclass of these has been discovered and linked to the Shimura and Shintani
lifts, as will first be described in this talk.  We then determine that these
functions are theta lifts of weak Maass forms, explaining many of the
properties previously discovered.  This is based upon joint work with Bringmann

Darmon points are a collection of conjectural generalizations of Heegner
points on modular elliptic curves. Algorithms for their effective
calculation are useful for gathering numerical evidence supporting their
conjectured properties. In addition, in some cases they can be used in
practice as very efficient methods for computing algebraic points in such
elliptic curves. In this talk I will recall two constructions of Darmon
points in different settings, and the algorithms of Darmon-Logan and
Darmon-Green-Pollack that allow for their computation in certain elliptic
curves.

We will study the asymptotics of the torsion in the cohomology
of certain arithmetic groups with coefficients in certain Z-modules
if the rank of the Z-modules tends to infinity. Our first main result is
that for cocompact groups the torsion grows exponentially where the
leading term is proportional to the volume of the underlying locally
symmetric space. Our method is based on studying the analytic torsion of
locally symmetric spaces with coefficients in flat vector bundles. If these space