| In number theory and arithmetic geometry, we expect an algebraic object such as a Mordell-Weil group of an abelian variety and an analytic object such as an L-function of an abelian variety to be inherently related. A classic example of this is the Birch and Swinnerton-Dyer (BSD) conjecture. The BSD conjecture predicts that the rank of the Mordell-Weil group of an elliptic curve is equal to the order of zero at s = 1 of the L-function of the same elliptic curve. Its modulo 2 version is the parity conjecture, which would be good evidence for the BSD conjecture. In this paper, we prove the parity conjecture for the p-Selmer group when p is a good supersingular reduction prime. Using a similar idea, we also prove algebraic functional equations for the +/--Selmer groups defined by S. Kobayashi. |
