Growth of Selmer groups in generalized dihedral extensions
(Un exposé de Karl Rubin au séminaire de
théorie des nombres de Chevaleret le 26 juin 2006)
Résumé :
In joint work with Barry Mazur, we obtain lower bounds for
Selmer ranks of elliptic curves over dihedral extensions of number fields.
If F/k is a dihedral extension of number fields of degree 2n with n odd,
and E is an elliptic curve over k that has odd rank over the quadratic
extension K of k in F, then standard conjectures (and a root number
calculation) predict that E(F) has rank at least n. The only case where
one can presently prove anything close to this bound is when K is
imaginary quadratic, and E(F) contains Heegner points.
Mazur and I prove unconditionally that if n is a power of an odd prime p,
F/K is unramified at all primes where E has bad reduction, all primes
above p split in K/k, and the p-Selmer corank of E/K is odd, then the
p-Selmer corank of E/F is at least n. This provides a large class of
examples of Z_p^d-extensions where the Selmer module is not a torsion
Iwasawa module.