Refined class number formulas and Kolyvagin systems

We use the theory of Kolyvagin systems to prove (most of) a refined class number formula conjectured by Darmon. We show that, for every odd prime p , each side of Darmon’s conjectured formula (indexed by positive integers n ) is ‘almost’ a p -adic Kolyvagin system as n varies. Using the fact that the space of Kolyvagin systems is free of rank one over ℤ p , we show that Darmon’s formula for arbitrary n follows from the case n =1, which in turn follows from classical formulas.