In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325):

Let $K$ be a number field and let $K^{cycl}$ be the cyclotomic extension of $K$ generated by all roots of unity. Let $E$ and $E'$ be two elliptic curves such that, over $\bar{K}$,

(i) $E$ and $E'$ have no complex multiplication;

(ii) The $l$-adic representations $(\rho_l)$, $(\rho'_l)$ attached to $E$ and $E'$ don't become isomorphic over any finite extension of $K$.

Then $K(E_{tors}) \cap K(E'_{tors})$ is finite over $K^{cycl}$.

My question is whether this holds for $E$ and $E'$ defined over a function field? If this hasn't already been considered somewhere with an argument specific to the function field case, then maybe a specialization argument would work. Could anyone please provide a reference where there are similar specialization arguments used, or a standard reference for the basic theory of these specialization theorems?

In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325):

Let $K$ be a number field and let $K^{cycl}$ be the cyclotomic extension of $K$ generated by all roots of unity. Let $E$ and $E'$ be two elliptic curves such that, over $\bar{K}$,

(i) $E$ and $E'$ have no complex multiplication;

(ii) The $l$-adic representations $(\rho_l)$, $(\rho'_l)$ attached to $E$ and $E'$ don't become isomorphic over any finite extension of $K$.

Then $K(E_{tors}) \cap K(E'_{tors})$ is finite over $K^{cycl}$.

My question is whether this holds for $E$ and $E'$ defined over a function field? If this hasn't already been considered somewhere with an argument specific to the function field case, then maybe a specialization argument might work? Could anyone please provide a reference where there are similar specialization arguments used, or a standard reference for the basic theory of these specialization theorems?

In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325):

Let $K$ be a number field and let $K^{cycl}$ be the cyclotomic extension of $K$ generated by all roots of unity. Let $E$ and $E'$ be two elliptic curves such that, over $\bar{K}$,

(i) $E$ and $E'$ have no complex multiplication;

(ii) The $l$-adic representations $(\rho_l)$, $(\rho'_l)$ attached to $E$ and $E'$ don't become isomorphic over any finite extension of $K$.

Then $K(E_{tors}) \cap K(E'_{tors})$ is finite over $K^{cycl}$.

My question is whether this holds for $E$ and $E'$ defined over a function field? If this hasn't already been considered somewhere with an argument specific to the (probably easier) function field case, then maybe a specialization argument would work. Could anyone please provide a reference where there are similar specialization arguments used, or a standard reference for the basic theory of these specialization theorems?

In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325):

Let $K$ be a number field and let $K^{cycl}$ be the cyclotomic extension of $K$ generated by all roots of unity. Let $E$ and $E'$ be two elliptic curves such that, over $\bar{K}$,

(i) $E$ and $E'$ have no complex multiplication;

(ii) The $l$-adic representations $(\rho_l)$, $(\rho'_l)$ attached to $E$ and $E'$ don't become isomorphic over any finite extension of $K$.

Then $K(E_{tors}) \cap K(E'_{tors})$ is finite over $K^{cycl}$.

My question is whether this holds for $E$ and $E'$ defined over a function field? If this hasn't already been considered somewhere with an argument specific to the function field case, then maybe a specialization argument would work. Could anyone please provide a reference where there are similar specialization arguments used, or a standard reference for the basic theory of these specialization theorems?

In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325):

Let $K$ be a number field and let $K^{cycl}$ be the cyclotomic extension of $K$ generated by all roots of unity. Let $E$ and $E'$ be two elliptic curves such that, over $\bar{K}$,

(i) $E$ and $E'$ have no complex multiplication;

(ii) The $l$-adic representations $(\rho_l)$, $(\rho'_l)$ attached to $E$ and $E'$ don't become isomorphic over any finite extension of $K$.

Then $K(E_{tors}) \cap K(E'_{tors})$ is finite over $K^{cycl}$.

My question is whether this holds for $E$ and $E'$ defined over a function field?

In Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 259--331 (1972), Serre proved the following (Theorem 6 ′′, p. 325):

Let $K$ be a number field and let $K^{cycl}$ be the cyclotomic extension of $K$ generated by all roots of unity. Let $E$ and $E'$ be two elliptic curves such that, over $\bar{K}$,

(i) $E$ and $E'$ have no complex multiplication;

(ii) The $l$-adic representations $(\rho_l)$, $(\rho'_l)$ attached to $E$ and $E'$ don't become isomorphic over any finite extension of $K$.

Then $K(E_{tors}) \cap K(E'_{tors})$ is finite over $K^{cycl}$.

My question is whether this holds for $E$ and $E'$ defined over a function field? If this hasn't already been considered somewhere with an argument specific to the (probably easier) function field case, then maybe a specialization argument would work. Could anyone please provide a reference where there are similar specialization arguments used, or a standard reference for the basic theory of these specialization theorems?