# Refined Class Number Formulas and Kolyvagin Systems

 Title: Refined Class Number Formulas and Kolyvagin Systems Author: Rubin, Karl; Mazur, Barry C. Note: Order does not necessarily reflect citation order of authors. Citation: Mazur, Barry, and Karl Rubin. 2011. Refined class number formulas and Kolyvagin systems. Compositio Mathematica 147(1): 56-74. Full Text & Related Files: 0909.3916v1.pdf (273.1Kb; PDF) 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. Published Version: doi://10.1112/S0010437X1000494X Other Sources: http://arxiv.org/abs/0909.3916v1 Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10076145 Downloads of this work: