Alice Pozzi, University of Bristol

PDL C38
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 SwinnertonDyer Conjecture in rank 1.
In recent years, Darmon and collaborators have proposed a (conjectural) theory of "real multiplication" relying on padic methods. In particular, conjectural analogues of Heegner points and singular moduli for real quadratic fields have been obtained exploiting the geometry of the Drinfeld padic upper half plane.
In this talk, I will discuss aspects of the theory of complex multiplication and compare them with their analogues for real quadratic fields. I will then present a result confirming the conjectural properties of singular moduli in a special case.