Compositio Mathematica

Table of Contents - January 2011 - Volume 147, Issue 01  

Research Article

Refined class number formulas and Kolyvagin systems

Barry Mazura1 and Karl Rubina2

a1 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (email: mazur@math.harvard.edu)

a2 Department of Mathematics, UC Irvine, Irvine, CA 92697, USA (email: krubin@math.uci.edu)

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 xs2124p, 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

  • 11R42 (primary);
  • 11R27;
  • 11R29;
  • 11R37 (secondary)

Footnotes

This material is based upon work supported by the National Science Foundation under grants DMS-0700580 and DMS-0757807.