Gross–Prasad-assisted large-prime Selmer vanishing conjecture for symmetric powers

Let $F$ be a totally real number field, let $r$ be a positive integer, and let $A$ be an elliptic curve over $F$ with no complex multiplication over $\ovl F$. Suppose there exist a self-dual automorphic representation $\Pi$ of $\operatorname{GL}_{2r+1}(\mbf A_F)$ of the stated cuspidal or isobaric-sum type, quadratic spaces $\mbf V\subset\mbf V_\sharp=\mbf V\oplus Fe$ with $\dim\mbf V=2r+1$ and $e$ of norm $1$, cuspidal representations $\pi_0$ and $\pi_1$ with the stated Arthur parameters, and cusp forms $f_0\in\pi_0$ and $f_1\in\pi_1$. Let $\iota:\bx O(\mbf V)\hookrightarrow\bx O(\mbf V_\sharp)$ be induced by the inclusion, and define

$$\mcl P_{\bx{GP}}(f_0,f_1)=\int_{\bx O(\mbf V)(F)\backslash\bx O(\mbf V)(\Ade_F)}f_0(h)f_1(\iota(h))\,\bx dh.$$

Assume this orthogonal Gross–Prasad period is nonzero, and let $E$ be a strong coefficient field of $\Pi$. Gross–Prasad-assisted Selmer vanishing conjecture. There exists an effective constant $N(F,A,r)$ depending only on $F$, $A$, and $r$, such that

$$\bx H_f^1\left(F,\operatorname{Sym}^{2r-1}\etH^1(A_{\ovl F};\bb Q_\ell)(r)\right)=0$$

for all rational primes $\ell>N(F,A,r)$ underlying a preadmissible place $\lbd$ of $E$ with respect to $(A,\Pi)$. The conjecture links nonvanishing orthogonal Gross–Prasad periods with vanishing of Bloch–Kato Selmer groups for sufficiently large primes; its status is not resolved in the supplied source.