Elliptic curves and Hilbert's 12th problem |
Abstract: The theory of complex multiplication allows the explicit analytic construction of algebraic points on elliptic curves, points which are defined over abelian extensions of imaginary quadratic fields. Until recently, this theory supplied the only known method (aside from direct search) for constructing points on elliptic curves.
I will introduce the theory of complex multiplication from a concrete, computational point of view and outline a conjectural method for constructing points defined over abelian extensions of real quadratic fields. This method is analytic, relying on a mixture of complex and p-adic integration. It suggests the possibility of analytic constructions of global points on elliptic curves going beyond the classical framework. |
Web page: Alexandru I. Suciu | Created: September 6, 2000 | |
Comments to: alexsuciu@neu.edu | URL: http://www.math.neu.edu/bhmn/darmon.html |