Let $F$ be a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction.

Let $\gamma:H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/im(\kappa_v)$ where $\Sigma$ is the collection of the primes above p, the bad primes and arch primes and the product runs over the primes of $\Sigma$

If $Sel_{p^\infty}(E/K)$ is finite, i.e E is of rank 0 (assuming sha is finite), then Cassel's theorem says that $coker(\gamma)\cong E(F)_p$. In particular, if $E(F)[p]=0$, the map is surjective.

Is anything similar true for positive rank? The only reference for this that I know is Greenberg's notes where he proves that the map will be surjective over $F_\infty$ (the cycl $\mathbb Z_p$ extension), for any rank (assuming the selmer is $\Lambda-$cotorsion, and without assuming anything on the torsion). Are there any conditions in which over number fields we can still say the map is surjective for any rank?

Let $F$ be a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction.

Let $\gamma:H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/\operatorname{im}(\kappa_v)$ where $\Sigma$ is the collection of the primes above $p,$ the bad primes and arch primes and the product runs over the primes of $\Sigma$

If $\operatorname{Sel}_{p^\infty}(E/K)$ is finite, i.e E is of rank 0 (assuming sha is finite), then Cassel's theorem says that $\operatorname{coker}(\gamma)\cong E(F)_p$. In particular, if $E(F)[p]=0$, the map is surjective.

Is anything similar true for positive rank? The only reference for this that I know is Greenberg's notes where he proves that the map will be surjective over $F_\infty$ (the cycl $\mathbb Z_p$ extension), for any rank (assuming the selmer is $\Lambda$-cotorsion, and without assuming anything on the torsion). Are there any conditions in which over number fields we can still say the map is surjective for any rank?

Let $F$ be a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction.

Let $\gamma:H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/im(\kappa_v)$ where $\Sigma$ is the collection of the primes above p, the bad primes and arch primes and the product runs over the primes of $\Sigma$

If $E$ is of rank 0, then Cassel's theorem says that $coker(\gamma)\cong E(F)_p$. In particular, if $E(F)[p]=0$, the map is surjective.

Is anything similar true for positive rank? The only reference for this that I know is Greenberg's notes where he proves that the map will be surjective over $F_\infty$ (the cycl $\mathbb Z_p$ extension), for any rank (assuming the selmer is $\Lambda-$cotorsion, and without assuming anything on the torsion). Are there any conditions in which over number fields we can still say the map is surjective for any rank?

Let $F$ be a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction.

Let $\gamma:H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/im(\kappa_v)$ where $\Sigma$ is the collection of the primes above p, the bad primes and arch primes and the product runs over the primes of $\Sigma$

If $Sel_{p^\infty}(E/K)$ is finite, i.e E is of rank 0 (assuming sha is finite), then Cassel's theorem says that $coker(\gamma)\cong E(F)_p$. In particular, if $E(F)[p]=0$, the map is surjective.

Is anything similar true for positive rank? The only reference for this that I know is Greenberg's notes where he proves that the map will be surjective over $F_\infty$ (the cycl $\mathbb Z_p$ extension), for any rank (assuming the selmer is $\Lambda-$cotorsion, and without assuming anything on the torsion). Are there any conditions in which over number fields we can still say the map is surjective for any rank?

Let $F$ a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction.

Let $\gamma:H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/im(\kappa_v)$ where $\Sigma$ is the collection of the primes above p, the bad primes and arch primes and the product runs over the primes of $\Sigma$

If $E$ is of rank 0, then Cassel's theorem says that $coker(\gamma)\cong E(F)_p$. In particular, if $E(F)[p]=0$, the map is surjective.

Is anything similar true for positive rank? The only reference for this that I know is Greenberg's notes where he proves that the map will be surjective over $F_\infty$ (the cycl $\mathbb Z_p$ extension), for any rank (assuming the selmer is $\Lambda-$cotorsion, and without assuming anything on the torsion). Are there any conditions in which over finite fields we can still say the map is surjective for any rank?

Let $F$ be a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction.

Let $\gamma:H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/im(\kappa_v)$ where $\Sigma$ is the collection of the primes above p, the bad primes and arch primes and the product runs over the primes of $\Sigma$

If $E$ is of rank 0, then Cassel's theorem says that $coker(\gamma)\cong E(F)_p$. In particular, if $E(F)[p]=0$, the map is surjective.

Is anything similar true for positive rank? The only reference for this that I know is Greenberg's notes where he proves that the map will be surjective over $F_\infty$ (the cycl $\mathbb Z_p$ extension), for any rank (assuming the selmer is $\Lambda-$cotorsion, and without assuming anything on the torsion). Are there any conditions in which over number fields we can still say the map is surjective for any rank?