Let $K$ be a quadratic number field. Let $E$ be an elliptic curve defined over $\mathbb{Q}$.

Let $\mathrm{Sha}(E/K)$ denote the Tate-Shafarevich group of $E/K$. Can we explicitly write down the norm map of the Tate-Shafarevich group $\mathrm{Sha}(E/K) \to \mathrm{Sha}(E/\mathbb{Q})$?

(Reference: What's the Hilbert class field of an elliptic curve?)

At first, I thought that the map $[C] \mapsto [C] + [C]^{\sigma}$ would provide the desired map. However, it does not commute with the Galois action, meaning that $([C] + [D])^{\sigma} \neq [C]^{\sigma} + [D]^{\sigma}$ in general.

Thank you for your help.

Let $K$ be a quadratic number field. Let $\sigma$ be a generator of Galois group of $K/\Bbb{Q}$. Let $E$ be an elliptic curve defined over $\mathbb{Q}$.

Let $\mathrm{Sha}(E/K)$ denote the Tate-Shafarevich group of $E/K$. Can we explicitly write down the norm map of the Tate-Shafarevich group $\mathrm{Sha}(E/K) \to \mathrm{Sha}(E/\mathbb{Q})$?

(Reference: What's the Hilbert class field of an elliptic curve?)

At first, I thought that the map $[C] \mapsto [C] + [C]^{\sigma}$ would provide the desired map. However, it does not commute with the Galois action, meaning that $([C] + [C]^{\sigma} )^{\sigma} \neq [C]^{\sigma} + [C]$ in general, thus this is not well-defined.

Thank you for your help.

Let $K$ be a quadratic number field.Let $E$ be an elliptic curve defined over $\Bbb{Q}$.

$Sha(E/K)$ be Tate-Shafarevich group of $E/K$. Can we explicitly write down norm map of Tate-Shafarevich group $Sha(E/K)\to Sha(E/\Bbb{Q})$ ?

(cf. What's the Hilbert class field of an elliptic curve?)

I first thought that the map $[C]\to [C]+[C]^{\sigma}$ gives the map. But it does not commute with Galois action, that is, $([C]+[D])^{\sigma}=[C]^{\sigma}+[D]^{\sigma}$ does not hold in general.

Thank you for your help.

Let $K$ be a quadratic number field. Let $E$ be an elliptic curve defined over $\mathbb{Q}$.

Let $\mathrm{Sha}(E/K)$ denote the Tate-Shafarevich group of $E/K$. Can we explicitly write down the norm map of the Tate-Shafarevich group $\mathrm{Sha}(E/K) \to \mathrm{Sha}(E/\mathbb{Q})$?

(Reference: What's the Hilbert class field of an elliptic curve?)

At first, I thought that the map $[C] \mapsto [C] + [C]^{\sigma}$ would provide the desired map. However, it does not commute with the Galois action, meaning that $([C] + [D])^{\sigma} \neq [C]^{\sigma} + [D]^{\sigma}$ in general.

Thank you for your help.