891 research outputs found
-Selmer growth in extensions of degree
There is a known analogy between growth questions for class groups and for
Selmer groups. If is a prime, then the -torsion of the ideal class group
grows unboundedly in -extensions of a fixed number
field , so one expects the same for the -Selmer group of a nonzero
abelian variety over . This Selmer group analogue is known in special cases
and we prove it in general, along with a version for arbitrary global fields.Comment: 19 pages; final version, to appear in Journal of the London
Mathematical Societ
On the asymptotic growth of Bloch-Kato-Shafarevich-Tate groups of modular forms over cyclotomic extensions
We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of
a modular form f over the cyclotomic Zp-extension of Q under the assumption
that f is non-ordinary at p. In particular, we give upper bounds of these
groups in terms of Iwasawa invariants of Selmer groups defined using p-adic
Hodge Theory. These bounds have the same form as the formulae of Kobayashi,
Kurihara and Sprung for supersingular elliptic curves.Comment: To appear in Canad. J. Mat
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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