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.
Keywords
class number one problem, real quadratic fields,
Stark–Heegner points, rigid analytic cocycles