The Riemann Theta Function

Abstract

Let Λ be a lattice in ℂg, i.e. a subgroup of ℂg which is discrete and of rank 2g; the quotient M = ℂg/Λ is a compact complex manifold, called a complex torus. Let L be a holomorphic line bundle on M, and π: ℂg → M the projection. A well-known theorem in complex analysis asserts that any holomorphic line (or even vector) bundle on ℂg is holomorphically trivial. Let \(h:{\pi ^*}(L) \to {{\Bbb C}^9} \times {\Bbb C}\) be a trivialisation. If λ ∈ Λ and z ∈ ℂg, then the isomorphisms \({\pi ^*}{(L)_z} \to {\Bbb C}{\text{and }}{\pi ^*}{(L)_{z + \lambda }} \to {\Bbb C}\) differ by multiplication by a constant since \({\pi ^*}{(L)_z} = {\pi ^*}{(L)_{z + \lambda }} = {L_{\pi (z)}};\) if we denote this constant by φλ(z), then for λ ∈ Λ, z →φλ(z) is a holomorphic function without zeros, and we have, for λ, μ ∈ Λ, $$ {\varphi _\mu }\left( {z + \lambda } \right){\varphi _\lambda }\left( z \right) = {\varphi _{\lambda + \mu }}\left( z \right), z \in {\mathbb{C}^9}. $$