This question is from Silverman's 'the arithmetic of elliptic curves',$p134$.
Let $K$ be a field of characteristic $p > 0$, let $E_1/ K$ and $ E_2/K$ be elliptic curves, and let $\phi : E_1 \to E_2$ be a nonzero isogeny defined over $K$. Further, let $f: \hat{E_1} \to \hat{E_2} $ be the homomorphism of formal groups induced by $\phi$.
My question:
How does an isogeny $\phi$ on elliptic curves induces a homomorphism of corresponding formal groups?
I guess $f(T)＝\phi(T)$,but I cannot check this is actually homomorphism.
My question is, I would like to know the confirming process
$\phi(F_1(x,y))＝F_2(\phi(x),\phi(y))$,where $F_1$ and $F_2$ are formal group law of elliptic curve $E_1$, $E_2$.
If you have an elliptic curve $E_j/k:y_j^2=x_j^3+a_j x+b_j$ then
the group law is given by a rational function in 4 variables $x_j,y_j,x_j',y_j'$ which can be rewritten as a rational function in $x_j/y_j,1/y_j,x_j'/y_j',1/y_j'$.
$1/y_j$ is in $k[[x_j/y_j]]$
whence the group law is given by a formal series $F_j\in k[[x_j/y_j, x_j'/y_j']]$,
the power series expansion of the algebraic-rational function giving $x_j(A+B)/y_j(A+B)$ from $x_j(A)/y_j(A)$ and $x_j(B)/y_j(B)$. This is the formal group law. When $k=\Bbb{C}$ this power series is an analytic function.
Your isogeny $f:E_1\to E_2$ is given by a pair of rational functions in $x_1,y_1$, can be rewritten as a pair of rational functions in the $x_1/y_1,1/y_1$ coordinates. Replacing $1/y_1$ by its formal series $\in k[[x_1/y_1]$ this pair of rational functions is given by one formal series $\phi(x_1/y_1)\in k[[x_1/y_1]]$, the power series expansion of the algebraic-rational function giving the $x_2/y_2$ coordinate from $x_1/y_1$.
$T\to \phi(T)$ is your homomorphism of formal group law.
We have just made a change of coordinate and embedded the rational-algebraic functions defining the group law and the isogeny in some larger ring of formal series. The group law and the isogeny commute, so do their formal series representation.