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 $\mathbf{Z}_p$, we show that Darmon's formula for arbitrary $n$ follows from the case $n=1$, which in turn follows from classical formulas.