When $E$ is an elliptic curve over a number field $L$, with complex multiplication by a maximal order in an imaginary quadratic field $K$, Silverman’s Advanced Topics in the Arithmetic of Elliptic Curves, chapter 2, section 9 explains how to associate a Grossencharakter of $K$ to $E$. Is it possible to associate a Grossencharakter or Galois representation of $K$ to $E$, if instead we take $E$ to be an elliptic curve over a field $F$ of positive characteristic with CM by $K$? If so, and $E$ happens to be the base change to $F$ of the reduction of some elliptic curve over a number field, is there a relationship between the Grossencharakter of the curve over the number field and the analogous object associated to $E$?

I should add that I’m specifically interested in the case where $F$ is a function field.

When $E$ is an elliptic curve over a number field $L$, with complex multiplication by a maximal order in an imaginary quadratic field $K$, Silverman’s Advanced Topics in the Arithmetic of Elliptic Curves, chapter 2, section 9 explains how to associate a Grössencharakter of $K$ to $E$. Is it possible to associate a Grössencharakter or Galois representation of $K$ to $E$, if instead we take $E$ to be an elliptic curve over a field $F$ of positive characteristic with CM by $K$? If so, and $E$ happens to be the base change to $F$ of the reduction of some elliptic curve over a number field, is there a relationship between the Grössencharakter of the curve over the number field and the analogous object associated to $E$?

I should add that I’m specifically interested in the case where $F$ is a function field.

When $E$ is an elliptic curve over a number field $L$, with complex multiplication by a maximal order in an imaginary quadratic field $K$, Silverman’s Advanced Topics in the Arithmetic of Elliptic Curves, chapter 2, section 9 explains how to associate a Grossencharakter of $K$ to $E$. Is it possible to associate a Grossencharakter or Galois representation of $K$ to $E$, if instead we take $E$ to be an elliptic curve over a field $F$ of positive characteristic with CM by $K$? If so, and $E$ happens to be the base change to $F$ of the reduction of some elliptic curve over a number field, is there a relationship between the Grossencharakter of the curve over the number field and the analogous object associated to $E$?

When $E$ is an elliptic curve over a number field $L$, with complex multiplication by a maximal order in an imaginary quadratic field $K$, Silverman’s Advanced Topics in the Arithmetic of Elliptic Curves, chapter 2, section 9 explains how to associate a Grossencharakter of $K$ to $E$. Is it possible to associate a Grossencharakter or Galois representation of $K$ to $E$, if instead we take $E$ to be an elliptic curve over a field $F$ of positive characteristic with CM by $K$? If so, and $E$ happens to be the base change to $F$ of the reduction of some elliptic curve over a number field, is there a relationship between the Grossencharakter of the curve over the number field and the analogous object associated to $E$?

I should add that I’m specifically interested in the case where $F$ is a function field.

When $E$ is an elliptic curve over a number field $L$ with complex multiplication by a maximal order in an imaginary quadratic field $K$, Silverman’s Advanced Topics in the Arithmetic of Elliptic Curves, chapter 2, section 9 explains how to associate a Grossencharakter of $K$ to $E$. Is it possible to associate a Grossencharakter or Galois representation of $K$ to $E$, if instead we take $E$ to be an elliptic curve over a field $F$ of positive characteristic with CM by $K$? If so, and $E$ happens to be the base change to $F$ of the reduction of some elliptic curve over a number field, is there a relationship between the Grossencharakter of the curve over the number field and the analogous object associated to $E$?

When $E$ is an elliptic curve over a number field $L$, with complex multiplication by a maximal order in an imaginary quadratic field $K$, Silverman’s Advanced Topics in the Arithmetic of Elliptic Curves, chapter 2, section 9 explains how to associate a Grossencharakter of $K$ to $E$. Is it possible to associate a Grossencharakter or Galois representation of $K$ to $E$, if instead we take $E$ to be an elliptic curve over a field $F$ of positive characteristic with CM by $K$? If so, and $E$ happens to be the base change to $F$ of the reduction of some elliptic curve over a number field, is there a relationship between the Grossencharakter of the curve over the number field and the analogous object associated to $E$?