The determination by Heegner, Baker and Stark of the complete list of imaginary quadratic orders of class number one relies critically on the theory of complex multiplication. A conjectural extension of this theory to real quadratic fields based on the notion of rigid analytic elliptic cocycles is shown to yield similar lists for some explicit families of real quadratic orders with small regulators, for instance for discriminants of the form n2+4, n2−4 and 4n2 + 1.

Joint work with Henri Darmon.