Mazur, BarryRubin, Karl2012-12-192011Mazur, Barry, and Karl Rubin. 2011. Refined class number formulas and Kolyvagin systems. Compositio Mathematica 147(1): 56-74.1570-5846http://nrs.harvard.edu/urn-3:HUL.InstRepos:10076145We 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.en-USJournal Article2012-12-1910.1112/S0010437X1000494X