 Riemann theta function

For a symmetric matrix $B$ with positive definite imaginary part, this theta function converges normally on compact subsets of ${\mathbb{C}}^g$. It obeys $\theta (z+m+Bn,B)=e^{-\pi i{n}^{t}Bn-2\pi i{n}^{t}z}\theta (z,B)$, 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.