= Riemann theta function
{c}
{title2=$\theta(z,B)=\sum_{n\in\mathbb Z^g}e^{\pi i n^tBn+2\pi i n^tz}$}
= Riemann's theta function
{c}
{synonym}
For a symmetric matrix $B$ with positive definite imaginary part, this <theta function> converges normally on <compact> subsets of $\mathbb C^g$. It obeys $\theta(z+m+Bn,B)=e^{-\pi i n^tBn-2\pi i n^tz}\theta(z,B)$, so its zeros define a <divisor> on the associated <complex torus>. If $B$ is a curve's normalized period matrix, its zero <divisor> is a translate of the effective degree-$g-1$ locus, by the <Riemann vanishing theorem>. A general symmetric positive period matrix need not come from a curve.
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