Refined class number formulas and Kolyvagin systems

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 ${\mathbf{Z}}_p$, we show that Darmon's formula for arbitrary $n$ follows from the case $n=1$, which in turn follows from classical formulas.

References in corpus (1)

• Refined class number formulas and Kolyvagin systems

Cited by in corpus (8)

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• Refined abelian Stark conjectures and the equivariant leading term conjecture of Burns
• On higher special elements of $p$-adic representations
• On the theory of higher rank Euler, Kolyvagin and Stark systems, IV: the multiplicative group
• A generalization of Darmon's conjecture for Euler systems for general p-adic representations