Towards a theory of real multiplication

The theory of elliptic curves with complex multiplication has yielded some striking arithmetic applications. Classically, Kronecker tackled the problem of constructing abelian extensions of imaginary quadratic fields, known as the "Kronecker Jugendtraum", via singular moduli, values of modular functions at imaginary quadratic points of the complex upper half plane.  The theory of complex multiplication is also employed in the construction of Heegner points, global points on elliptic curves playing a crucial role in the proof of the Birch and Swinnerton-Dyer Conjecture in rank 1.

In recent years, Darmon and collaborators have proposed a (conjectural) theory of "real multiplication" relying on p-adic methods. In particular, conjectural analogues of Heegner points and singular moduli for real quadratic fields have been obtained exploiting the geometry of the Drinfeld p-adic upper half plane.