Linear system which gives $(m,n)$-polarization?

Question

What is the dimension of $H^0(T,\mathcal{L})$, where $T$ is a complex torus of dimension $2$ and $\mathcal{L}$ is a line bundle which gives $T$ a $(m,n)$-polarization?

Answer 1

This is the Riemann-Roch Theorem for abelian varieties (see Birkenhake-Lange, Section 3.6 for example). In general if you have a $(d_1,\ldots,d_g)$-polarization $\mathcal{L}$ then $$H^0(T,\mathcal{L})=d_1\cdots d_g.$$ In your case it would be $mn$.