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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: mazur@math.harvard.edu)
Karl Rubin
Affiliation:
Department of Mathematics, UC Irvine, Irvine, CA 92697, USA (email: krubin@math.uci.edu)
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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

References

[1]Darmon, H., Thaine’s method for circular units and a conjecture of Gross, Canad. J. Math. 47 (1995), 302–317.CrossRefGoogle Scholar
[2]Gross, B., On the values of abelian L-functions at s=0, J. Fac. Sci. Univ. Tokyo 35 (1988), 177–197.Google Scholar
[3]Hales, A., Stable augmentation quotients of abelian groups, Pacific J. Math. 118 (1985), 401–410.CrossRefGoogle Scholar
[4]Hayes, D., The refined 𝔭-adic abelian Stark conjecture in function fields, Invent. Math. 94 (1988), 505–527.CrossRefGoogle Scholar
[5]Mazur, B. and Rubin, K., Kolyvagin systems, Mem. Amer. Math. Soc. 799 (2004).Google Scholar
[6]Mazur, B. and Rubin, K., Introduction to Kolyvagin systems, in Stark’s conjectures: recent work and new directions, Contemporary Mathematics, vol. 358 (American Mathematical Society, Providence, RI, 2004), 207–221.CrossRefGoogle Scholar
[7]Mazur, B. and Tate, J., Refined conjectures of the ‘Birch and Swinnerton–Dyer type’, Duke Math. J. 54 (1987), 711–750.CrossRefGoogle Scholar
[8]Rubin, K., A Stark conjecture ‘over ℤ’ for abelian L-functions with multiple zeros, Ann. Inst. Fourier (Grenoble) 46 (1996), 33–62.CrossRefGoogle Scholar
[9]Rubin, K., Euler systems, Annals of Mathematics Studies, vol. 147 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
[10]Stark, H., L-functions at s=1. IV. First derivatives at s=0, Adv. Math. 35 (1980), 197–235.CrossRefGoogle Scholar