Tying together the analytic Jacobian and algebraic Jacobian

Abstract

So far in this Chapter, we have defined an algebraic variety Jac C and studied its invariant flows. In Chapter II, we associated to any compact Riemann Surface C a complex torus Jac C. If C is hyperelliptic so that both constructions apply, they are isomorphic by Abel’s theorem. We would now like to make this isomorphism explicitly, i.e., express the algebraic coordinates on Jac C-θ as theta functions.