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Refined class number formulas and Kolyvagin systems

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  17 August 2010

Barry Mazur  and
Karl Rubin
Barry Mazur
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (email: [email protected])
Karl Rubin
Affiliation:
Department of Mathematics, UC Irvine, Irvine, CA 92697, USA (email: [email protected])
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Abstract

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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 ℤp, we show that Darmon’s formula for arbitrary n follows from the case n=1, which in turn follows from classical formulas.

MSC classification

Secondary: 11R42: Zeta functions and $L$-functions of number fields 11R27: Units and factorization 11R29: Class numbers, class groups, discriminants 11R37: Class field theory
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 1 , January 2011 , pp. 56 - 74
Copyright
Copyright © Foundation Compositio Mathematica 2010

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