= Solution
Write $g_2=60G_4$ and $g_3=140G_6$, with the lattice sums taken over nonzero lattice elements. Reindexing the normally convergent series shows that the <Weierstrass elliptic function> $\wp$ is an <even function>, and consequently $\wp'$ is an <odd function>. Expanding $(z-w)^{-2}$ near zero and cancelling the odd lattice sums gives the <Laurent coefficients of the Weierstrass elliptic function>
$$
\wp(z)=z^{-2}+3G_4z^2+5G_6z^4+O(z^6),\qquad
\wp'(z)=-2z^{-3}+6G_4z+20G_6z^3+O(z^5).
$$
Thus the <elliptic function> $\wp'^2-4\wp^3+60G_4\wp$ has no pole at zero and its constant term is $-140G_6$. By periodicity it has no poles anywhere. An entire <elliptic function> is bounded on a fundamental parallelogram and hence everywhere, so <Liouville theorem> makes it constant. This proves the <Weierstrass elliptic differential equation>
$$
\boxed{\wp'^2=4\wp^3-g_2\wp-g_3.}
$$
We first verify that the cubic is smooth, rather than presupposing it when using its group law. There are three nonzero <two-torsion points of a complex torus> $h_j\in(\tfrac12\Lambda)/\Lambda$. Oddness and periodicity imply $\wp'(h_j)=0$. Since $\wp'$ has a triple pole and no other poles on the torus, <value multiplicity of an elliptic function> says that these three distinct zeros exhaust its zeros and are simple. Put $e_j=\wp(h_j)$. Each $\wp-e_j$ has a double zero at $h_j$, since $\wp'(h_j)=0$ and the zero of $\wp'$ is simple. If $e_i=e_j$ with $i\ne j$, this same function would have at least four zeros counted with <multiplicity>, although its only pole is double. Therefore the <half-period values of the Weierstrass elliptic function> are distinct. They are roots of $4x^3-g_2x-g_3$, so all its roots are distinct and the affine curve is a <smooth algebraic curve>. Its <projective closure> is
$$
Y^2Z=4X^3-g_2XZ^2-g_3Z^3.
$$
It has the single point $O=(0:1:0)$ at infinity. The derivative with respect to $Z$ of the defining homogeneous polynomial is nonzero at $O$, proving smoothness there too.
For any finite value $a$, <value multiplicity of an elliptic function> gives exactly two solutions of $\wp(z)=a$, counting <multiplicities>. Evenness supplies the pair $z,-z$. At a nonzero half-period these coincide and the zero is double; elsewhere they are distinct. An even <elliptic function> $f$ therefore descends to a meromorphic function of $x=\wp(z)$ on the <Riemann sphere>. To justify descent at a half-period, put $z=h+u$: both $f(h+u)$ and $\wp(h+u)$ are even in $u$, and $\wp(h+u)-\wp(h)$ is $u^2$ times a nonvanishing even analytic function. The local <Laurent series> of $f$ is consequently meromorphic in the coordinate $x-e$. At zero the same argument uses $1/x=z^2+O(z^6)$. Every <meromorphic function> on the <Riemann sphere> is a <rational function>, proving that <even elliptic functions are rational in the Weierstrass function>.
For an arbitrary <elliptic function> $f$, its even part $(f(z)+f(-z))/2$ is $A(\wp)$. Its odd part divided by $\wp'$ is an even <meromorphic function>, hence is $B(\wp)$ with $B$ rational. We have therefore proved the <elliptic function-field decomposition>
$$
\boxed{f(z)=A(\wp(z))+\wp'(z)B(\wp(z)),\qquad A,B\in\mathbb C(x).}
$$
Uniqueness follows by taking even and odd parts. In particular the field of <elliptic functions> is $\mathbb C(\wp,\wp')$.
Define $\Phi(0)=O$ and $\Phi(z)=(\wp(z),\wp'(z))$ elsewhere on $\mathbb C/\Lambda$. Periodicity makes the map well defined. Near zero, the projective chart $Y\ne0$ has coordinates $X/Y=\wp/\wp'=-z/2+O(z^5)$ and $Z/Y=1/\wp'=-z^3/2+O(z^7)$, so the extension is holomorphic. The fiber description above proves injectivity: the two candidates $z,-z$ are distinguished by $\wp'$ unless they coincide at a half-period. It also proves surjectivity, since every $x$ occurs and the two derivatives are the two allowed $y$ values, with $y=0$ at a branch value.
Finally, a nonvertical line $y=ax+b$ pulls back to the <elliptic function> $\wp'-a\wp-b$, whose only pole is triple at zero. Its three zeros $z_1,z_2,z_3$, counted with <intersection multiplicity>, satisfy $z_1+z_2+z_3=0$ modulo $\Lambda$ by the <zero-pole sum of an elliptic function>. On the cubic, reflection $y\mapsto-y$ corresponds to $z\mapsto-z$. The <chord-and-tangent group law> therefore gives $\Phi(z_1)+\Phi(z_2)=\Phi(-z_3)=\Phi(z_1+z_2)$. Vertical lines give inverse pairs; tangencies are included through repeated zeros, and the identity cases follow from the extension at zero. This completes the <Weierstrass uniformization by half-period values>: \b[$\Phi$ is a group isomorphism.]
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