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Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.

Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.

On the other hand, there exists the corestriction map $cores: H^1(H,M) \to H^1(G,M)$.

What is known about the kernel of $cores$? Are there any known results that explicitly describe the kernel or embed it into certain cohomology?

I am particularly interested in the following case: Let $K$ be a quadratic number field. Consider $G=Gal(\overline{K}/\Bbb{Q})$ and $H= Gal(\overline{K}/K)$. Let $M = E(\overline{K})$. For the restriction map, we can express $ker(res)$ as $H^1(Gal(K/\Bbb{Q}), E(K))$ as mentioned above. However, I am facing difficulty in controlling $ker(cores)$.

Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.

Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.

On the other hand, there exists the corestriction map $cores: H^1(H,M) \to H^1(G,M)$.

What is known about the kernel of $cores$? Are there any known results that explicitly describe the kernel or embed it into certain cohomology?

I am particularly interested in the following case: Let $K$ be a quadratic number field. Consider $G=Gal(\overline{K}/\Bbb{Q})$ and $H= Gal(\overline{K}/K)$. Let $E$ be an elliptic curve defined over $\Bbb{Q}$. Let $M = E(\overline{K})$. For the restriction map, we can express $ker(res)$ as $H^1(Gal(K/\Bbb{Q}), E(K))$ as mentioned above. However, I am facing difficulty in controlling $ker(cores)$.

Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.

Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.

On the other hand, there exists the corestriction map $cores: H^1(H,M) \to H^1(G,M)$.

What is known about the kernel of $cores$? Are there any known results that explicitly describe the kernel or embed it into certain cohomology?

I am particularly interested in the following case: Let $K$ be a number field. Consider $G=Gal(\overline{K}/\Bbb{Q})$ and $H= Gal(\overline{K}/K)$. Let $M = E(\overline{K})$. For the restriction map, we can express $ker(res)$ as $H^1(Gal(K/\Bbb{Q}), E(K))$ as mentioned above. However, I am facing difficulty in controlling $ker(cores)$.

Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.

Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.

On the other hand, there exists the corestriction map $cores: H^1(H,M) \to H^1(G,M)$.

What is known about the kernel of $cores$? Are there any known results that explicitly describe the kernel or embed it into certain cohomology?

I am particularly interested in the following case: Let $K$ be a quadratic number field. Consider $G=Gal(\overline{K}/\Bbb{Q})$ and $H= Gal(\overline{K}/K)$. Let $M = E(\overline{K})$. For the restriction map, we can express $ker(res)$ as $H^1(Gal(K/\Bbb{Q}), E(K))$ as mentioned above. However, I am facing difficulty in controlling $ker(cores)$.

Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.

Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.

On the other hand, there exists the corestriction map $cores: H^1(H,M) \to H^1(G,M)$.

What is known about the kernel of $cores$? Are there any known results that explicitly describe the kernel or embed it into certain cohomology?

I am particularly interested in the following case: Let $K$ be a number field. Consider $G=Gal(\overline{K}/\Bbb{Q})$ and $H= Gal(\overline{K}/K)$. Let $M = E(\overline{K})$. For the restriction map, we can express $ker(res)$ as $H^1(Gal(K/\Bbb{Q}), E(K))$ as mentioned above. However, I am facing difficulty in controlling $ker(cores)$.

Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.

Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.

On the other hand, there exists the corestriction map $cores: H^1(H,M) \to H^1(G,M)$.

What is known about the kernel of $cores$? Are there any known results that explicitly describe the kernel or embed it into certain cohomology?

I am particularly interested in the following case: Let $K$ be a number field. Consider $G=Gal(\overline{K}/\Bbb{Q})$ and $H= Gal(\overline{K}/K)$. Let $M = E(\overline{K})$. For the restriction map, we can express $ker(res)$ as $H^1(Gal(K/\Bbb{Q}), E(K))$ as mentioned above. However, I am facing difficulty in controlling $ker(cores)$.