For example, elliptic curve $E:y^2＝x^3-x$ can be transformed to move the identity element to the origin, $(z,w)=(0,0)$, we do the change of coordinates $z=-x/y$, $w=-1/y$. The equation of this curve then becomes $w=z^3+zw^2\quad$
My question is ; Does E':$w=z^3+zw^2 $have group structure, just like ' take two points from $E'$,and line through those points intersects at $E'$ , and connect the point with $（0,0）∈（z,w）$, then the line intersects with $E'$ again, and this is the sum of first two points' ?
The graph of $w=z^3+zw^2\quad$seems completely different in affine plane from usual $y^2＝x^3-x$, so I'm suspicious. But maybe we can say the group structure above holds because they are both same Weierstrass from in projective plane, so is isomorphic to Picard group of $E$.
We divide by $-y^3=(-y)^3$ both sides in $y^2 =x^3-x$.
Then we get $(-1/y)=(-x/y)^3 \color{red}{-} (-x/y)(-1/y)^2$, so the new curve has the affine equation $w=z^3-zw^2$.
For the homogenized equation, $$ \begin{aligned} &E'\ &:\ WT^2&=Z^3-ZW^2\ ,\qquad\text{associated to}\\ &E'_{\text{affine}}\ &:\ w&=z^3-zw^2\ , \end{aligned} $$ obtained using the correspondence $w=W/T$, $z=Z/T$, so that an affine point $(z,w)$ goes to $[Z:W:T]=[z:w:1]$.
There are three points "at infinity" on $E'$ (w.r.t. the chosen affine part), the ones with $T=0$: They are $\Omega_a=[a:1:0]$ with $a$ among $-1$, $0$, $1$. They correspond to the three points $(a,0)$ on $E$.
We will work in the sequel over $\Bbb Q$.
The line $Z=0$ intersects the curve $E'$ in three points, $O:=[0:0:1]=(0,0)$ with multiplicity one and $\Omega$ with multiplicity two. This should be considered when writing formulas. Below, i will consider only the "generic case", which is the case when all constructed points involved are $\ne \Omega$.
We will describe the operations which makes a posteriori $O\in E'(\Bbb Q)$ to be the neutral element of the group $(E'(\Bbb Q),+)$.
Consider two $\Bbb Q$-rational affine points, $P_1=(z_1,w_1)$, $P_2=(z_2,w_2)$. The equation of the line $P_1P_2$ is then of the shape $$ (L)\ :\ w = mz+n\ , $$ with $m=\displaystyle\frac{w_1-w_2}{z_1-z_2}$, which makes sense for power series purposes around $O=(0,0)$ since $w$ has order $3$, $z$ order $1$ around zero, and unfortunately we will also need $n=\displaystyle\frac{z_1w_2-z_2w_1}{z_1-z_2}$. In case the two points conicide, $P_1=P_2$, the line has the "same" equation, but $m,n$ are obtained by formally differentiating $(E')$, to get $$dw =3z^2\; dz-w^2\; dz-2zw\; dw\ .$$ Formally replacing $(z,w)$ by $(z_1,w_1)$, and $dz, dw$ by $(z-z_1)$ and respectively $(w-w_1)$ we obtain the first order Taylor polynomial approximation around $P_1$. This gives $(1+2z_1w_1)\; dw= (3z_1^2-w_1^2)\; dz$. So $\displaystyle m=\frac {3z_1^2-w_1^2}{1+2z_1w_1}$, which is also fine around $O$. Then set $n$ again so that $w_1=mz_1+n$.
The system with two equations, $(E')$ and $(L)$, in the two unknowns has three solutions, Bezout, we obtain an equation of degree three in $z$ by the linear substitution of $w$ from $(L)$ in $(E')$, thus getting: $$ 0 = z^3-z(mz+n)^2-(mz+n)\ . $$
Now we consider the line $OR_3'$, which intersects $E'$ in a third point $R_3$. This point is the point we need. Now note that together with a point $(z,w)$ on $(E')$ the point $(-z,-w)$ is also a point on $(E')$. With this observation we can directly accept the formula for $R_3=(z_3,w_3):=(-z_3',-w_3')$: $$ \begin{aligned} z_3 &=z_1+z_2-\frac{2mn}{1-m^2}\ ,\\ w_3 &=mz_3-n\ . \end{aligned} $$
To have some power series experiment, i will use sage, and some point with coordinates in the power series ring $R=\Bbb Q[[ q ]]$. If this feels alien while regarding the target in the OP, please ignore the following. I insert this since some power series application was suggested in the comments.
Investigations on this path have their origin in
Arithmetic of Elliptic Curves, John T. Tate, §3, Expansion near Zero, The Formal Group.
With a completely different starting point and target, rather for my purposes, there is still some open project in my bottom drawer related to the following experiment, which uses power series coming from the modular parametrization...
I will use sage in the following.
The addition and multiplication of "blind tuples", corresponding to affine points on the given curve, is implemented as follows.
Now we can start experimenting.
We may start with $z=q$, and ask for the corresponding lift of $w$ in the power series ring in $q$. Since in our special case $w$ is the solution of the equation of second degree $qw^2+w-q^3=0$, we already have a formula for a starting point. We will compute some multiples of this point.
This gives:
We can check that the above points are on the curve. For instance for the last point...
The experiment using power series related to the modular parametrization are of the shape
This gives: