(Uniformisation)
Let \(\Lambda \subseteq \CC\) be a lattice and \(E_\Lambda \da V(y^2 = 4x^3 - g_2 x - g_3) \subseteq \PP^2_{/\CC}\). Then \(\Delta = g_2^3 - 27g_3^2 \neq 0\), so \(E_\Lambda\) is a smooth cubic, and \[ \phi \colon \CC/\Lambda \to E_\Lambda, \qquad z \mapsto \tv{\wp(z) : \wp'(z) : 1}, \quad 0 \mapsto \tv{0:1:0} , \] is an isomorphism of Riemann surfaces which is also a group isomorphism onto \((E_\Lambda, p_0)\) with \(p_0 = \phi(0)\). Conversely every elliptic curve over \(\CC\) is isomorphic to some \(E_\Lambda\): given \(c_2, c_3\) with \(c_2^3 - 27c_3^2 \neq 0\) there is a lattice with \(g_2 = c_2\) and \(g_3 = c_3\).
(Abel's theorem on the torus)
A divisor \(\sum n_i (a_i)\) on \(\CC/\Lambda\) is the divisor of an elliptic function exactly when \[ \sum n_i = 0 \qquad\text{and}\qquad \sum n_i a_i = 0 \text{ in } \CC/\Lambda . \]
The two halves of the first statement are proved by opposite methods and it is worth keeping them apart.
That \(\phi\) lands in the cubic is the differential equation and nothing else. That it is bijective is degree counting: \(\wp\) has degree \(2\) on the torus, \(\wp'\) degree \(3\), and \(\wp\) is even, so \(\wp(z) = \wp(z')\) forces \(z' = \pm z\) and the sign is then pinned by the odd function \(\wp'\); the point at infinity is hit once, by \(z=0\). Holomorphic and bijective gives an isomorphism of Riemann surfaces, and GAGA upgrades it to an isomorphism of algebraic curves.
Surjectivity of \(\Lambda \mapsto (g_2,g_3)\) is the one step that is not formal, and it is really a statement about the modular function: \(J = g_2^3/\Delta\) takes every complex value, and the scaling \(\Lambda \mapsto \alpha\Lambda\) sends \((g_2,g_3) \mapsto (\alpha^{-4}g_2, \alpha^{-6}g_3)\), which moves \((g_2,g_3)\) anywhere on the fibre of \(J\) over a fixed value. So: \(J\) surjective supplies the shape, homothety supplies the scale.
The second theorem is why the analytic and the algebraic group laws agree, and it is the exact analogue of Abel’s theorem for a general curve. The condition \(\sum n_i = 0\) is degree zero; the condition \(\sum n_i a_i = 0\) is the Abel–Jacobi map, which for genus \(1\) is the identity \(\CC/\Lambda \to \CC/\Lambda\). Specialising to \(a_1 + a_2 - b - 0\): three points of the torus sum to \(0\) exactly when the corresponding points of \(E_\Lambda\) are cut out by a line, which is the chord-and-tangent law. The torsion subgroup is immediate from the lattice model: \(E[n] = \tfrac{1}{n}\Lambda / \Lambda \cong (\ZZ/n)^2\), generated by \(\tfrac{a}{n} + \tfrac{b}{n}\tau\), and topologically \(E \cong (\RR/\ZZ)^2\).
(Normalisations)
The model \(y^2 = 4x^3 - g_2x - g_3\) is what \(\wp\) produces; the Legendre model \(y^2 = x(x-1)(x-\lambda)\) is what the branch-point normalisation produces. Passing between them is a linear change of variable together with \(x \mapsto \alpha^2 x\), \(y \mapsto \alpha^3 y\), which is exactly the homothety \(\Lambda \mapsto \alpha \Lambda\) seen on the curve. Any statement that a formula for \(j\) or \(g_2\) “is” a number depends on which model is in force, so say which one before computing.
Dependencies
D-CRVWEIERElliptic functions, the Weierstrass $\wp$-function, and $(\wp')^2 = 4\wp^3 - g_2\wp - g_3$uses