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Let $A$ be an abelian variety on $\mathbb{Q}_p$ with supersingular reduction and $\mathbb{Q}_p(\mu_{p^{\infty}})/\mathbb{Q}_p$ the cyclotomic extension.

Do we have : $A(\mathbb{Q}_p (\mu_{p^{\infty}}))[p^\infty] = 0$ ?

This is true for elliptic curves : Kobayashi, «Iwasawa theory for elliptic curves at supersingular primes» Prop. 8.7 .

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