= Weierstrass uniformization by half-period values
{c}
{title2=$\mathbb C/\Lambda\cong\{y^2=4x^3-g_2x-g_3\}\cup\{O\}$}
The <Weierstrass elliptic function> and its derivative give an isomorphism $z\mapsto(\wp(z),\wp'(z))$ from a one-dimensional <complex torus> to a <smooth plane cubic>, with $0$ sent to its identity at infinity. The three nonzero <two-torsion points of a complex torus> are zeros of $\wp'$. Its triple pole implies that these three zeros are simple. If two <half-period values of the Weierstrass elliptic function> coincided, $\wp-e$ would have at least four zeros counted with <multiplicity>, contradicting its double pole. The three cubic roots are therefore distinct, proving <smoothness of an algebraic variety>. The degree-two map $\wp$ has fibers $\{z,-z\}$, distinguished by $\wp'$ away from the half-periods, proving bijectivity. A line pulls back to an <elliptic function> with a triple pole, and the <zero-pole sum of an elliptic function> says that its three intersections sum to zero on the torus. This gives exactly the <chord-and-tangent group law>, including repeated intersections interpreted by <intersection multiplicity>.
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