Let $X$ be a closed compact Riemann surface of genus $g$. Then I can get $\operatorname{Jac}(X)$, Jacobian of the $X$ by Abel-Jacobi mapping. $\operatorname{Jac}(X)=C^g/\Lambda$ admits non-trivial $\theta$ function(i.e. a function defined on $C^g$ surely vanishes somewhere or equivalently, $\operatorname{Jac}(X)$ admits non-trivial meromorphic section) iff $\Lambda\otimes_RR\cong C^g$ admits a positive semi-definite Riemann form.

$\textbf{Q:}$ How do I get this Riemann form from lattice $\Lambda$? It seems I could not see any direct prescription without constructing $\theta$ function. For $g=1$, it is clear. For $g>1$, it is very unclear why I should even have a positive semi-definite Riemann form.

$\textbf{Q:}$ In general there is no reason to expect a tori having meromoprhic section. When tori admits meromorphic section and suppose one knows only it admits a meromorphic section, do I even know this riemann form is determined by lattice?


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