We use the theory of continued fractions in conjunction with ideal theory (often called the infrastructure) in real quadratic fields to give new class number 2 criteria and link this to a canonical norm-induced quadratic polynomial. By doing so, this provides a real quadratic field analogue of the well-known result by Hendy (1974) for complex quadratic fields. We illustrate with several example...

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...

Acknowledgements I am undoubtedly indebted to my two supervisors, Alf van der Poorten and Hugh Williams. Most students would feel lucky to have one good supervisor whereas I had the privilege of having two excellent mathematicians to guide me. Their help, wisdom and infinite patience has been invaluable, and I am very grateful for it. The Mathematics Department at Macquarie University has provi...