Refined class number formulas and Kolyvagin systems

TL;DR

This paper proves a refined class number formula conjectured by Darmon using Kolyvagin systems, showing that the formula holds for all positive integers by reducing it to the classical case.

Contribution

It introduces a novel approach connecting Darmon's conjecture with the theory of Kolyvagin systems, establishing the formula's validity for all cases.

Findings

Darmon's refined class number formula is valid for all positive integers n.

The space of Kolyvagin systems is free of rank one over Z_p.

The proof reduces the general case to the classical case n=1.

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.