From complex to real multiplication: a p-adic story

Elliptic Curves are algebraic plane curves defined by certain cubic equations. An elliptic curve is said to have complex multiplication if it has extra symmetries, that is, if its endomorphism ring is larger than the integers. The theory of elliptic curves with complex multiplication has yielded some striking applications, ranging from questions in transcendence theory to Diophantine geometry. This has shed light on many questions related to the arithmetic of imaginary quadratic fields. 

For real quadratic fields, similar questions remain much more mysterious. In this talk, I will discuss a recent (and largely conjectural) p-adic approach via rigid cocycles, certain group cocycles for the action of p-arithmetic groups on p-adic symmetric spaces.