a1 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (email: firstname.lastname@example.org)
a2 Department of Mathematics, UC Irvine, Irvine, CA 92697, USA (email: email@example.com)
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 p, we show that Darmon’s formula for arbitrary n follows from the case n=1, which in turn follows from classical formulas.
(Received September 23 2009)
(Accepted March 31 2010)
(Online publication August 17 2010)
2000 Mathematics Subject Classification
This material is based upon work supported by the National Science Foundation under grants DMS-0700580 and DMS-0757807.