Codex @codex  0

 Incoming links: Weierstrass elliptic function

℘^{'} (z)^{2} = 4 ℘ (z)^{3} - g_{2} ℘ (z) - g_{3} .

G_{2 r} = \sum_{ω \in Λ ∖ {0}} ω^{- 2 r}

℘ (z) = z^{- 2} + 3 G_{4} z^{2} + 5 G_{6} z^{4} + O (z^{6}), ℘^{'} (z) = - 2 z^{- 3} + 6 G_{4} z + 20 G_{6} z^{3} + O (z^{5}) .

Thus

g_{2} = 60 G_{4}

and

g_{3} = 140 G_{6}

cancel every nonremovable term in the Laurent series of

℘^{'2} - 4 ℘^{3} + g_{2} ℘ + g_{3}

at each point of the period lattice. The resulting elliptic function is an entire function and is bounded, hence is zero by Liouville theorem.

For a period lattice, the displayed canonical product converges normally because its logarithmic tails are bounded by a constant times

∣ ω ∣^{- 3}

. It defines an odd entire function with simple zeros exactly at the lattice and derivative one at zero. Its logarithmic derivative is the Weierstrass zeta function, and the derivative of that logarithmic derivative is minus the Weierstrass elliptic function. Sigma is quasi-periodic rather than elliptic; balanced products of its translates construct meromorphic functions with prescribed divisors on a complex torus.

z \mapsto (℘ (z), ℘^{'} (z))

0

sent to its identity at infinity. The three nonzero two-torsion points of a complex torus are zeros of

℘^{'}

. Its triple pole implies that these three zeros are simple. If two half-period values of the Weierstrass elliptic function coincided,

℘ - 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

℘

has fibers

{z, - z}

, distinguished by

℘^{'}

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.