This question is related to this one. Tate module of CM elliptic curves

There seem to be several versions of "complex multiplication". Fact 1: We say $E/\mathbb{C}$ has CM if $End_C(E) \supsetneq Z$. It is equivalent that $End_C(E)$ is an order in a quadratic field.

Fact 2: Now suppose $E$ defined over $\it{ANY}$ field $K$, we say that $E$ has CM $\it over$ $\it K$ if $End_K(E) \supsetneq Z$. By the method Pete pointed out in the comment of the above MO question, if can be shown that in this case, the Galois group $Gal(\bar{K}/K)$ action on the Tate module $T_{\ell}(E)$ for $\ell \neq char(K)$ is abelian.

Question 1: Suppose $E$ defined over $\it{ANY}$ field $K$ with $ch K = 0$, if $E$ has CM $\it over$ $\it K$, is it true that $End_K(E)$ is an order in a quadratic field? And what about $End_{\bar{K}}(E)$, is it an order in a quadratic field? We require $ch K =0$ because when $K$ is a finite field, $End_{\bar{K}}(E)$ is always bigger than $Z$ and could be an order in a quaternion algebra.

QUestion 2: Now suppose $E$ defined over a $\it number$ $\it field$ $K$, thus it could be seen as an elliptic curve over $C$ as well.Is it true that $End_K(E) \supsetneq Z$ if and only if $End_C(E) \supsetneq Z$?

This question is related to this one. Tate module of CM elliptic curves

There seem to be several versions of "complex multiplication". Fact 1: We say $E/\mathbb{C}$ has CM if $End_C(E) \supsetneq Z$. It is equivalent that $End_C(E)$ is an order in a quadratic field.

Fact 2: Now suppose $E$ defined over $\it{ANY}$ field $K$, we say that $E$ has CM $\it over$ $\it K$ if $End_K(E) \supsetneq Z$. By the method Pete pointed out in the comment of the above MO question, if can be shown that in this case, the Galois group $Gal(\bar{K}/K)$ action on the Tate module $T_{\ell}(E)$ for $\ell \neq char(K)$ is abelian.

Question 1: Suppose $E$ defined over $\it{ANY}$ field $K$ with $ch K = 0$, if $E$ has CM $\it over$ $\it K$, is it true that $End_K(E)$ is an order in a quadratic field? And what about $End_{\bar{K}}(E)$, is it an order in a quadratic field? We require $ch K =0$ because when $K$ is a finite field, $End_{\bar{K}}(E)$ is always bigger than $Z$ and could be an order in a quaternion algebra.

QUestion 2: Now suppose $E$ defined over a $\it number$ $\it field$ $K$, thus it could be seen as an elliptic curve over $C$ as well.Is it true that $End_K(E) \supsetneq Z$ if and only if $End_C(E) \supsetneq Z$?

This question is related to this one. Tate module of CM elliptic curves

There seem to be several versions of "complex multiplication". Fact 1: We say $E/\mathbb{C}$ has CM if $End_C(E) \supsetneq Z$. It is equivalent that $End_C(E)$ is an order in a quadratic field.

Fact 2: Now suppose $E$ defined over $\it{ANY}$ field $K$, we say that $E$ has CM $\it over$ $\it K$ if $End_K(E) \supsetneq Z$. By the method Pete pointed out in the comment of the above MO question, if can be shown that in this case, the Galois group $Gal(\bar{K}/K)$ action on the Tate module $T_{\ell}(E)$ for $\ell \neq char(K)$ is abelian.

Question 1: Suppose $E$ defined over $\it{ANY}$ field $K$ with $ch K \neq 0$, if $E$ has CM $\it over$ $\it K$, is it true that $End_K(E)$ is an order in a quadratic field? And what about $End_{\bar{K}}(E)$, is it an order in a quadratic field? We require $ch K \neq 0$ because when $K$ is a finite field, $End_{\bar{K}}(E)$ is always bigger than $Z$ and could be an order in a quaternion algebra.

QUestion 2: Now suppose $E$ defined over a $\it number$ $\it field$ $K$, thus it could be seen as an elliptic curve over $C$ as well.Is it true that $End_K(E) \supsetneq Z$ if and only if $End_C(E) \supsetneq Z$?

This question is related to this one. Tate module of CM elliptic curves

There seem to be several versions of "complex multiplication". Fact 1: We say $E/\mathbb{C}$ has CM if $End_C(E) \supsetneq Z$. It is equivalent that $End_C(E)$ is an order in a quadratic field.

Fact 2: Now suppose $E$ defined over $\it{ANY}$ field $K$, we say that $E$ has CM $\it over$ $\it K$ if $End_K(E) \supsetneq Z$. By the method Pete pointed out in the comment of the above MO question, if can be shown that in this case, the Galois group $Gal(\bar{K}/K)$ action on the Tate module $T_{\ell}(E)$ for $\ell \neq char(K)$ is abelian.

Question 1: Suppose $E$ defined over $\it{ANY}$ field $K$ with $ch K = 0$, if $E$ has CM $\it over$ $\it K$, is it true that $End_K(E)$ is an order in a quadratic field? And what about $End_{\bar{K}}(E)$, is it an order in a quadratic field? We require $ch K =0$ because when $K$ is a finite field, $End_{\bar{K}}(E)$ is always bigger than $Z$ and could be an order in a quaternion algebra.

QUestion 2: Now suppose $E$ defined over a $\it number$ $\it field$ $K$, thus it could be seen as an elliptic curve over $C$ as well.Is it true that $End_K(E) \supsetneq Z$ if and only if $End_C(E) \supsetneq Z$?