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.

Suppose that an elliptic curve E over Q has good supersingular reduction at p. We prove that Kobayashi's plus/minus Selmer group of E over a Z p -extension has no proper Λ-submodule of finite index under some suitable conditions, where Λ is the Iwasawa algebra of the Galois group of the Z p -extension. This work is analogous to Greenberg's result in the ordinary reduction case.2010 Mathematics subject classification: primary 11G.

Abstract. We study the Iwasawa µ-and λ-invariants of the non-primitive plus/minus Selmer groups of elliptic curves for supersingular primes. We prove that they are constant for a family of elliptic curves with the same residual representation if the µ-invariant of any of them is 0. As an application we find a family of elliptic curves whose plus/minus Selmer groups have arbitrarily large λ-invariants.

In this paper we study the central values of L-functions associated to a large class of algebraic Hecke characters of imaginary quadratic fields. When these central values are nonzero, the Bloch-Kato conjecture predicts an exact formula for the algebraic parts of the central values in terms of periods and arithmetic data, most notably the Selmer groups corresponding to the Hecke characters. We investigate the nonvanishing of these central values, and prove the p-part of the Bloch-Kato conjecture in these cases for primes p which split in K .

We develop the plus/minus p-Selmer group theory and plus/minus padic L-function theory for an elliptic curve E with complex multiplication over an abelian extension F of the imaginary quadratic field K given by the complex multiplication of E when p is a prime inert over K/Q (i.e. supersingular). As a result, we prove that the characteristic ideal of the Pontryagin dual of the plus/minus p-Selmer group of E over the cyclotomic Zp-extension of F is generated by the plus/minus p-adic Lfunction of E. This work is a generalization of [12], [14], and [15].