The periods of the generalized Jacobian of a complex elliptic curve (Q487192)

Let \(\Lambda\) be the lattice generated by \((1,0)\), \((0,1)\) and \((\hat{\tau}, \tilde{\tau})\), with \(\hat{\tau}\not\in \mathbb{R}\). The authors are interested in determining the periods of the generalized Jacobian \(J_L\) of a complex elliptic curve \(\mathcal{C}\) with modulus \(L\). They show that the toroidal Lie group \(\mathcal{G}=\mathbb{C}^2/\Lambda\) is isomorphic to that generalized Jacobian \(J_L\). The proof is constructive, and it provides a relation between the Weierstrass sigma function and the elliptic function corresponding to the lattice in \(\mathbb{C}\) generated by \(1\) and \(\hat{\tau}\).