Appell–Humbert theorem

The Appell–Humbert theorem classifies holomorphic line bundles on a complex torus by a Riemann form together with a compatible semicharacter of its lattice.

For an integral alternating form $E$ arising as the imaginary part of a Hermitian form, a semicharacter satisfies $\alpha (\lambda +\mu )=e^{\pi iE(\lambda ,\mu )}\alpha (\lambda )\alpha (\mu )$. The Appell–Humbert theorem uses this datum together with the Hermitian form to specify a holomorphic line bundle on a complex torus. The ratio of two semicharacters is an ordinary unitary group homomorphism. On the square period lattice, with $H(z,w)=z\overline{w}$, all semicharacters are $\alpha (m+ni)=(-1{)}^{mn}{u}^m{v}^n$ for $u,v\in U(1)$.