### The parity conjecture for elliptic curves at supersingular reduction primes

Byoung Du (B. D.) Kim
2007 Compositio Mathematica
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. Introduction In number
more » ... ction In number theory and arithmetic geometry, we often expect an algebraic aspect and an analytic aspect to be closely related. A classic example is the Birch and Swinnerton-Dyer (BSD) conjecture. The BSD conjecture predicts a precise relation between the Mordell-Weil group and the L-function of an elliptic curve. Its statement is given in the following. Conjecture 1 (BSD conjecture). Suppose that E/Q is an elliptic curve defined over Q. Then we expect rankE(Q) = ord s=1 L /Q (E, s). Another conjecture closely related to the BSD conjecture is the BSD conjecture for a Selmer group. The Selmer group of an elliptic curve is a subgroup of a cohomology group associated to the torsion points of the elliptic curve, and the Shafarevich-Tate conjecture predicts that its corank is equal to the rank of E(Q). The BSD conjecture for a Selmer group is stated as follows. Conjecture 2 (BSD conjecture for a Selmer group). Let p be a prime number and E an elliptic curve defined over Q. We let Sel p (E/Q) denote the p-Selmer group of E over Q, then we expect We consider the modulo two version of the BSD conjecture for a Selmer group, namely the parity conjecture. Conjecture 3 (Parity conjecture). Let p be a prime number and E an elliptic curve defined over Q. We expect corank Zp Sel p (E/Q) ≡ ord s=1 L /Q (E, s) (mod 2). Note that this conjecture depends on the prime number p (as does Conjecture 2). Although we focus on this conjecture throughout this paper, we can state the BSD conjecture and the parity