Refined Class Number Formulas and Kolyvagin Systems

Abstract

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.