Stark-Heegner points for totally real fields

by Prof. Amod Agashe (Florida State University, USA)

Thursday 15 Dec 2011, 11:30 → 12:30 Asia/Kolkata

AG-77 (Colaba Campus)

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.