Refined class number formulas and Kolyvagin systems

Details Refined class number formulas and Kolyvagin systems Refined class number formulas and Kolyvagin systems

2009-09-22

arxiv.org/abs/0909.3916v1

Mathematics

Number Theory

Scientific paper

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.

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