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Let $E$ be an elliptic curve over $\mathbb{Q}$, with good reduction at $p$, and let $V = H^1_{et}(\overline{E}, \mathbb{Q}_p)$ be (the dual of) its (rationalized) Tate module.

Let $S^nV$ denote its $n$'th symmetric power, and let $H^1_f(\mathbb{Q}, S^nV)$ denote its Bloch-Kato Selmer group.

Are there any known unconditional results which give an upper bound on the dimension of the $BK$-Selmer group of $S^nV$ (or of its Dirichlet/Tate twists)?

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    $\begingroup$ When $n=2j$ is even, the Tate twist $S^{2j}(V)(j)$ has weight zero and is self-dual, and it is known that its BK Selmer group vanishes if $E$ is non-CM by Theorem 5.6 here (arxiv.org/pdf/1912.11265). For most other twists, this problem is very hard and there is some extensive literature on the cases $n=2$ and (more recently) $n=3$. BTW, no need to assume good reduction at $p$. $\endgroup$
    – Satan's Minion
    Commented Sep 12 at 5:23
  • $\begingroup$ This is literally what I was searching for, thanks. $\endgroup$
    – kindasorta
    Commented Sep 15 at 7:44

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