The symmetric structure of the plus/minus Selmer groups of elliptic curves over totally real fields and the parity conjecture

By improving the techniques of [B.D. Kim, The parity conjecture for elliptic curves at supersingular reduction primes, Compos. Math. 143 (2007) 47–72] we prove some symmetric structure of the minus Selmer groups of elliptic curves for supersingular primes. This structure was already known for the Selmer groups for ordinary primes [J. Nekovar, On the parity of ranks of Selmer groups II, C. R. Math. Acad. Sci. Paris Ser. I 332 (2) (2001) 99–104; J. Nekovar, Selmer complexes, Astérisque 310 (2006)]. One consequence is the parity conjecture over a totally real field under some conditions.