Isogenous elliptic curve.

Question

I'm studying elliptic curves and I have a question

Take two $k$-isogenous elliptic curves defined over a number field $k$ and fix a place $v$ of good reduction.

Are the reduced curves $\mathrm{mod} \:v$ isogenous?

Answer 1

Yes, they are. In fact, note that if $E,E'$ are $k$-isogenous, then their $l$-adic Tate modules are isomorphic as $Gal(\overline{k}/k)$-modules, where $l$ is any prime number not lying below $v$. Since $v$ is a place of good reduction, the Tate modules of the reduced curves are isomorphic as $Gal(\overline{\mathbb F_v}/\mathbb F_v)$-modules. Now a theorem of Tate (see for example Silverman, III.7.7) tells you that the map $$\hom (E_1,E_2)\otimes\mathbb Z_l\to \hom (T_l(E_1),T_l(E_2))$$ is an isomorphism when $E_1$, $E_2$ are elliptic curves over a finite field, and this proves your claim.