Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.

Consider the natural map $$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{Gal}(L_w/K_v), E(L_{v_L})), $$ where $\operatorname{Gal}(L_w/K_v)$ denotes the local Galois group corresponding to the place $v$.

For an abelian group $M$, let $M^*$ denote its Pontryagin dual.

In this context, the Tate-cohomology group is represented as $$ \hat{H}^0(\operatorname{Gal}(L/K),E(L)) = E(K)/(1+\sigma)E(L). $$(In other words, if we define a map $trace:E(L)\to E(K), P\to P+P^{\sigma}$, it is $ Coker(trace)$.)

According to p214 of the cited paper (link), it's claimed that $$ ({cokerF})^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $$ but this is asserted without further explanation.

Could you provide some insights or suggest strategies for proving this isomorphism?

N.B.

The linked paper reads the dual of diagram $(3)$ in p214 is the bottom diagram in p214. lem5 in p214 follows immediately if we confirm the titled isomorphism.

The notation used in this question is somewhat different from that of the paper. However, this question is still self-contained. I have provided a link to the paper for reference, but there is no mention of the isomorphism discussed in this question in the paper, even though it's described using slightly different notation.

Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.

Consider the natural map $$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{Gal}(L_w/K_v), E(L_{v_L})), $$ where $\operatorname{Gal}(L_w/K_v)$ denotes the local Galois group corresponding to the place $v$.

For an abelian group $M$, let $M^*$ denote its Pontryagin dual.

In this context, the Tate-cohomology group is represented as $$ \hat{H}^0(\operatorname{Gal}(L/K),E(L)) = E(K)/(1+\sigma)E(L). $$(In other words, if we define a map $\operatorname{trace}:E(L)\to E(K), P\to P+P^\sigma$, it is $ \operatorname{coker}(\operatorname{trace})$.)

According to p214 of the cited paper (link), it's claimed that $$ (\operatorname{coker} F)^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $$ but this is asserted without further explanation.

Could you provide some insights or suggest strategies for proving this isomorphism?

N.B.

The linked paper reads the dual of diagram $(3)$ on p 214 is the bottom diagram on p 214. lem 5 on p 214 follows immediately if we confirm the titled isomorphism.

The notation used in this question is somewhat different from that of the paper. However, this question is still self-contained. I have provided a link to the paper for reference, but there is no mention of the isomorphism discussed in this question in the paper, even though it's described using slightly different notation.

Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.

Consider the natural map $$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{Gal}(L_w/K_v), E(L_{v_L})), $$ where $\operatorname{Gal}(L_w/K_v)$ denotes the local Galois group corresponding to the place $v$.

For an abelian group $M$, let $M^*$ denote its Pontryagin dual.

In this context, the Tate-cohomology group is represented as $$ \hat{H}^0(\operatorname{Gal}(L/K),E(L)) = E(K)/(1+\sigma)E(L). $$(In other words, if we define a map $trace:E(L)\to E(K), P\to P+P^{\sigma}$, it is $ Coker(trace)$.)

According to p214 of the cited paper (link), it's claimed that $$ {cokerF}^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $$ but this is asserted without further explanation.

Could you provide some insights or suggest strategies for proving this isomorphism?

N.B.

The linked paper reads the dual of diagram $(3)$ in p214 is the bottom diagram in p214. lem5 in p214 follows immediately if we confirm the titled isomorphism.

The notation used in this question is somewhat different from that of the paper. However, this question is still self-contained. I have provided a link to the paper for reference, but there is no mention of the isomorphism discussed in this question in the paper, even though it's described using slightly different notation.

Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.

Consider the natural map $$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{Gal}(L_w/K_v), E(L_{v_L})), $$ where $\operatorname{Gal}(L_w/K_v)$ denotes the local Galois group corresponding to the place $v$.

For an abelian group $M$, let $M^*$ denote its Pontryagin dual.

In this context, the Tate-cohomology group is represented as $$ \hat{H}^0(\operatorname{Gal}(L/K),E(L)) = E(K)/(1+\sigma)E(L). $$(In other words, if we define a map $trace:E(L)\to E(K), P\to P+P^{\sigma}$, it is $ Coker(trace)$.)

According to p214 of the cited paper (link), it's claimed that $$ ({cokerF})^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $$ but this is asserted without further explanation.

Could you provide some insights or suggest strategies for proving this isomorphism?

N.B.

The linked paper reads the dual of diagram $(3)$ in p214 is the bottom diagram in p214. lem5 in p214 follows immediately if we confirm the titled isomorphism.

The notation used in this question is somewhat different from that of the paper. However, this question is still self-contained. I have provided a link to the paper for reference, but there is no mention of the isomorphism discussed in this question in the paper, even though it's described using slightly different notation.

Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.

Consider the natural map $$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{Gal}(L_w/K_v), E(L_{v_L})), $$ where $\operatorname{Gal}(L_w/K_v)$ denotes the local Galois group corresponding to the place $v$.

For an abelian group $M$, let $M^*$ denote its Pontryagin dual.

In this context, the Tate-cohomology group is represented as $$ \hat{H}^0(\operatorname{Gal}(L/K),E(L)) = E(K)/(1+\sigma)E(L). $$(In other words, if we define a map $trace:E(L)\to E(K), P\to P+P^{\sigma}$, it is $ Coker(trace)$.)

According to p214 of the cited paper (link), it's claimed that $$ {cokerF}^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $$ but this is asserted without further explanation.

Could you provide some insights or suggest strategies for proving this isomorphism?

N.B. The linked paper reads the dual of diagram $(3)$ in p214 is the bottom diagram in p214. lem5 in p214 follows immediately if we confirm the titled isomorphism.

Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.

Consider the natural map $$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{Gal}(L_w/K_v), E(L_{v_L})), $$ where $\operatorname{Gal}(L_w/K_v)$ denotes the local Galois group corresponding to the place $v$.

For an abelian group $M$, let $M^*$ denote its Pontryagin dual.

In this context, the Tate-cohomology group is represented as $$ \hat{H}^0(\operatorname{Gal}(L/K),E(L)) = E(K)/(1+\sigma)E(L). $$(In other words, if we define a map $trace:E(L)\to E(K), P\to P+P^{\sigma}$, it is $ Coker(trace)$.)

According to p214 of the cited paper (link), it's claimed that $$ {cokerF}^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $$ but this is asserted without further explanation.

Could you provide some insights or suggest strategies for proving this isomorphism?

N.B.

The linked paper reads the dual of diagram $(3)$ in p214 is the bottom diagram in p214. lem5 in p214 follows immediately if we confirm the titled isomorphism.

The notation used in this question is somewhat different from that of the paper. However, this question is still self-contained. I have provided a link to the paper for reference, but there is no mention of the isomorphism discussed in this question in the paper, even though it's described using slightly different notation.