Complex uniformization and lattices

A lattice is Z omega-one plus Z omega-two with the ratio of the periods nonreal. The quotient C/Λ is a compact one-dimensional complex Lie group with identity the zero class. Choosing a different integral basis with determinant one and the same orientation does not change Λ; it changes the displayed fundamental parallelogram. Choosing a nonzero complex scalar u produces a homothetic lattice uΛ and a holomorphically isomorphic torus via z mapped to uz.

The Weierstrass function is an even meromorphic lattice-periodic function with a double pole at each lattice point. Its derivative is odd with triple poles. The defining sum is regularized by subtracting the constant inverse-square contribution for each nonzero lattice point, and converges normally away from the lattice. Summing unmodified inverse squares in an arbitrary order would not be the same justified construction. The associated absolutely convergent Eisenstein sums define the cubic's coefficients.

The identity for the square of the derivative is wp-prime squared equal to four wp cubed minus g₂ wp minus g₃. Therefore the short Weierstrass coordinates are x=wp and y=wp-prime divided by two. Their equation has coefficients A=−g₂/4 and B=−g₃/4, and the zero class maps to O through the poles. The discriminant is nonzero for a genuine lattice. Using wp-prime itself as y without changing the cubic would introduce a factor-of-four error.

The invariant differential dx/(2y) pulls back to dz, and the map respects addition. Conversely, every elliptic curve over C admits such a uniformization as a complex analytic group. This theorem links analytic periods to algebraic geometry; it does not assert that arbitrary real periods recover a curve defined over Q. A rational model can have transcendental periods, and the lattice description alone does not decide which of its complex points are rational over a smaller field.

A holomorphic group homomorphism between two such tori lifts to z mapped to alpha z with alphaΛ contained in Λ-prime. If alpha is nonzero, it is an isogeny and its degree is the finite index of alphaΛ inside Λ-prime. The kernel is alpha-inverse Λ-prime modulo Λ. In characteristic zero these geometric kernel points already account for the degree, matching the algebraic separability statement. Multiplication by n has index n² because it scales both lattice directions.

A real-linear map on the underlying two-dimensional plane need not commute with multiplication by i, and therefore need not define a holomorphic elliptic-curve morphism. Likewise, a matrix changing a chosen lattice basis is not automatically an additional complex endomorphism. Extra endomorphisms require a complex number alpha preserving the lattice beyond the integer scalars. The square lattice provides an example through multiplication by i; a generic lattice has only integer endomorphisms.