= Semicharacter of a complex lattice
{title2=$\alpha:\Lambda\to U(1)$}
= Semicharacters of a complex lattice
{synonym}
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 $u,v\in U(1)$.
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