Stark-Heegner points for totally real fields

The classical theory of complex multiplication predicts the
existence of certain points called Heegner points defined over
quadratic imaginary fields on
elliptic curves (the curves themselves are defined over the rational
numbers). Henri Darmon observed that under
certain conditions, the Birch and Swinnerton-Dyer conjecture predicts
the existence of points of infinite order defined over real quadratic
fields on elliptic curves, and under such conditions, came up with a
conjectural construction of such points, which he called Stark-Heegner
points. Later, he and others extended this construction to many other
number fields. We will give a general construction of Stark-Heegner
points defined over quadratic extensions of totally real fields
(subject to some
restrictions); this is joint work with Mak Trifkovic. This
construction uses (in particular) theorems of Matsushima-Shimura and
Harder on the cohomology of arithmetic groups associated to totally
real fields, and in order to
generalize our construction to quadratic extensions of arbitrary
number fields, we seek analogs of these results for arbitrary number
fields, which we will mention in our talk.