Refined class number formulas for elliptic units

Abstract

We generalize the notions of Kolyvagin and pre-Kolyvagin systems to prove “refined class number formulas” for quadratic extensions of a quadratic imaginary fields $$K$$ of class number one. Our main result generalises the results and conjectures of Darmon (Canad. J. Math. 47:302–317, 1995), by replacing circular units in abelian extensions of $$\mathbb {Q}$$ by elliptic units in abelian extensions of K.