Compositio Mathematica

The parity conjecture for elliptic curves at supersingular reduction primes

Byoung Du (B. D.) Kim a1
a1 Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada

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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.

Maths Classification