For an elliptic curve E over a number field K, various
"modulo 2" versions of the Birch-Swinnerton-Dyer Conjecture
relate the parities of the Mordell-Weil rank, analytic rank and
The aim of this talk is to prove that over Q the parity of
every p-Selmer rank agrees with the parity of the analytic rank
(completing earlier work by Greenberg, Guo, Monsky,
Nekovar and Kim).
A key ingredient is a "local-to-global" expression for
various combinations of ranks of E over extensions of K,
which is motivated by Artin formalism for L-functions.
(This is joint work with Tim Dokchitser.)