Special Joint IAS/Princeton University Number Theory Seminar

In joint work with Barry Mazur, we obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields, giving evidence for a generalized Parity Conjecture. Heegner points can account for large Selmer groups in dihedral extensions of the rationals, but in the case of dihedral extensions of general number fields the source of the Selmer classes is a mystery.