Legendre Monodromy

For a given complex number λ not equal to 0 or 1, the elliptic curve described by the equation y^2 = x(x - 1)(x - λ) can be constructed as a complex manifold from the complex plane by making a branch-cut between 0 and 1, a branch cut between λ and infinity, and then glueing in a second copy of the complex plane along these cuts (this constructs the Riemann surface where there is a single-valued square root of x(x - 1)(x - λ), i.e. y). The branch-cuts here are illustrated in the left plane with red and purple dotted lines, while the green and orange solid curves give a basis for the integral homology of the curve (the green curve must doubled in the second copy of the complex plane). We can integrate the canonical differential dx/y along this homology basis as the contour integral of 1/sqrt(x(x - 1)(x - λ)) (or twice this integral in the case of the green curve). The values of the two period integrals are displayed in the right-hand side along with the lattice they generate; in particular, the elliptic curve corresponding to λ (with its point at infinity) is isomorphic to the complex numbers modulo the displayed lattice.

This app lets you drag around the point λ and will deform the contours to avoid the branch cuts as you do so. TIf you drag λ in a loop around 0 or 1, you will obtain a non-trivial monodromy showing how the basis vectors for this lattice are acted on by the fundamental group of the complex numbers minus 0 and 1. More on the mathematics of the Legendre family surrounding these constructions can be found, e.g., in Section 1.1 of Period Mappings and Period Domains by Carlson, Muller-Stach, and Peters.