On order of vanishing of characteristic elements

Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field with either good ordinary reduction or multiplicative reduction at each prime of $F$ above $p$. We shall study the characteristic element of the Selmer group of $E$ over a $p$-adic Lie extension. In particular, we relate the order of vanishing of these characteristic element evaluated at Artin representations to the Selmer coranks and their twists in the intermediate subextensions of the $p$-adic Lie extension.

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