On the algebraic functional equation for the mixed signed Selmer group over multiple Z p \mathbb {Z}_{p} -extensions

Let EE be an elliptic curve defined over a number field with good reduction at all primes above a fixed odd prime pp, where at least one of which is a supersingular prime of EE. In this paper, we will establish the algebraic functional equation for the mixed signed Selmer groups of EE over a multiple Zp\mathbb {Z}_{p}-extension.