The periods of the generalized Jacobian of a complex elliptic curve

Abstract

We show that the toroidal Lie group G = C^2/L, where L is the lattice generated by (1, 0), (0, 1) and (t, s), with t not in R, is isomorphic to the generalized Jacobian J_L of the complex elliptic curve E with modulus (1, t), defined by any divisor class D ≡ (M) + (N) of E ful lling M − N = [℘(s) : ℘'(s) : 1] in E. This follows from an apparently new relation between the Weierstrass sigma and elliptic function