How to confirm $\phi(F_1(x,y))＝F_2(\phi(x),\phi(y))$,where $F_1$ and $F_2$ are formal group law of elliptic curve $E_1$, $E_2$. This question is from Silverman's 'the arithmetic of elliptic curves',$p134$.
Let $K$ be a field of characteristic $p &gt; 0$, 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.
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.