Principality criterion on a complex elliptic curve ($\sum n_P[P]\text{ principal}⟺\sum n_P=0\text{ and }\sum n_{P}P=0\text{ in }E$)

For $E=\mathbb{C}/\Lambda$, integrate the logarithmic derivative of an elliptic function and its product with $z$ around a fundamental cell. The first integral forces total degree zero; the second forces the weighted point sum to lie in the lattice. Conversely choose lifts of the points with equal zero and pole sums, and form the balanced ratio of translated Weierstrass sigma functions. Their quasi-period factors cancel, giving an elliptic function with exactly the prescribed divisor. Separate zero and pole divisors require disjoint support; common terms cancel when only their difference is prescribed.