The parity conjecture for elliptic curves at supersingular reduction primes

Abstract

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.

The algebraic functional equation of an elliptic curve at supersingular primes

“…This is precisely [4] propositions 3.13 and 3.15 since k ∞ /k is a totally ramified extension. You can also see [4] propositions 3.17 and 3.18.…”

“…Let K ∞ be the cyclotomic Z p -extension of K. We define Sel − p (E/K ∞ ) following [5], [2], and [4]. We will explain this construction in the following sections.…”

“…The construction of plus norm subgroups in [4] does not seem to always work unlike minus norm subgroups. However, when K = Q(µ p ) or p splits completely over K/Q, we have the plus norm subgroups as constructed in [5] and [2], and we can prove the algebraic functional equation for the plus Selmer groups without modifying our technique.…”

The Iwasawa Invariants of the Plus/Minus Selmer Groups

“…Thus the kernel of ι ± is trivial, and the cokernel of ι ± is also trivial if the kernel of l∈Σ g ± l is trivial. It is proven in the proof of [5] is the biggest p-power divisor of the Tamagawa number c l . The Tamagawa number c l is 1 for all primes except l = 2, for which the Tamagawa number is 4.…”

On the Birch-Swinnerton-Dyer quotients modulo squares

“…To our best knowledge, over number fields Theorem 1.3 is the first general result of this kind, except for the work [7], [11] on curves with a p-isogeny. In contrast, the p-parity conjecture over ‫ޑ‬ was known in almost all cases, thanks to Birch, Stephens, Greenberg and Guo [3], [15], [16] (E CM), Kramer, Monsky [22], [26] (p D 2), Nekovář [28] (p potentially ordinary or potentially multiplicative) and Kim [18] (p supersingular). The results for Selmer groups in dihedral and false Tate curve extensions are similar to those recently obtained by Mazur-Rubin [23] and Coates-Fukaya-Kato-Sujatha [7], [8] Finally, we will need a slight modification of c.E=K/.…”