Grössencharakter or Galois representation associated to a CM elliptic curve in characteristic $p$

Question

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.

Answer 1

The construction of the Galois representation works equally well over every field. It is the Galois action on the $\ell$-adic Tate module, viewed as a rank one free module over $\mathbb Q_\ell \otimes_\mathbb Q K$.

On the other hand, a Grossencharacter over an arbitrary field of characteristic $p$ is not defined.

Given a curve $E$ over a number field $L$ with good reduction at a prime $v$ (i.e. that admits a smooth projective model over the local ring $\mathcal O_{L_v}$, it follows for $v\nmid \ell$ that the inertia group of $L$ at $v$ acts trivially on this Galois representation and hence the action of the decomposition group factors through the Galois group of the residue field $\kappa_v$, giving a Galois representation of the Galois group of $\kappa_v$. This produces the Galois representation of the reduction of $E$ at $v$. (This follows from the fact that for each $n$ the $\ell^n$-torsion of the $\mathcal O_{L_v}$-module of $E$ is finite étale over $\mathcal O_{L_v}$.)

To get from the Grossencharacter of $L$ to the character of the Galois group of $\kappa_v$, one just has to apply the correspondence between Grossencharacters and locally algebraic one-dimensional Galois representations and then apply this reduction process. This ends up with the Frobenius of $\kappa_v$ acting the same as the idele given by the uniformizer at $v$ and $1$ at every other place (or its inverse, depending on conventions).