The parity conjecture for elliptic curves at supersingular reduction primes
In number theory, the Birch and Swinnerton-Dyer (BSD) conjecture for a Selmer group relates the corank of a Selmer group of an elliptic curve over a number field to the order of zero of the associated $L$-function $L(E, s)$ at $s=1$. We study its modulo two version called the parity conjecture. The parity conjecture when a prime number $p$ is a good ordinary reduction prime was proven by Nekovar. We prove it when a prime number $p>3$ is a good supersingular reduction prime.(Published Online January 19 2007)
(Received August 1 2005)
(Accepted August 2 2006)
Key Words: elliptic curve; BSD conjecture; Iwasawa theory.