The periods of the generalized Jacobian of a complex elliptic curve

摘要

We show that the toroidal Lie group g = C-2/Lambda, where Lambda is the lattice generated by (1, 0), (0, 1) and ((tau) over cap(tau) over bar), with (tau) over cap is not an element of R, is isomorphic to the generalized Jacobian J L of the complex elliptic curve C with modulus (tau) over cap, defined by any divisor class L equivalent to (M) + (N) of C fulfilling M - N = [p((tau) over tilde) : p'((tau) over tilde) : 1] epsilon C. This follows from an apparently new relation between the Weierstrass sigma and elliptic functions.