Refined class number formulas and Kolyvagin systems

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 s...

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Bibliographic Details
Published inCompositio mathematica Vol. 147; no. 1; pp. 56 - 74
Main Authors Mazur, Barry, Rubin, Karl
Format Journal Article
LanguageEnglish
Published London, UK London Mathematical Society 01.01.2011
Cambridge University Press
Subjects
Integers
Mathematical analysis
Mathematics
11R42 (primary)
11R29
11R37 (secondary)
11R27
Online AccessGet full text
ISSN0010-437X
1570-5846
DOI10.1112/S0010437X1000494X

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Summary: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.
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ISSN:0010-437X
1570-5846
DOI:10.1112/S0010437X1000494X