587 research outputs found
Refined class number formulas and Kolyvagin systems
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 ,
each side of Darmon's conjectured formula (indexed by positive integers ) is
"almost" a -adic Kolyvagin system as varies. Using the fact that the
space of Kolyvagin systems is free of rank one over , we show
that Darmon's formula for arbitrary follows from the case , which in
turn follows from classical formulas
Finding large Selmer rank via an arithmetic theory of local constants
We obtain lower bounds for Selmer ranks of elliptic curves over dihedral
extensions of number fields.
Suppose is a quadratic extension of number fields, is an elliptic
curve defined over , and is an odd prime. Let denote the maximal
abelian -extension of that is unramified at all primes where has bad
reduction and that is Galois over with dihedral Galois group (i.e., the
generator of acts on by -1). We prove (under mild
hypotheses on ) that if the rank of the pro- Selmer group is
odd, then the rank of is at least for every finite extension
of in .Comment: Revised and improved. To appear in Annals of Mathematic
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