A pre-modular form Zn(σ;τ) of weight 21n(n+1) is
introduced for each n∈N, where (σ,τ)∈C×H, such that for Eτ=C/(Z+Zτ), every
non-trivial zero of Zn(σ;τ), namely σ∈Eτ[2],
corresponds to a (scaling family of) solution to the mean field equation
\begin{equation} \tag{MFE} \triangle u + e^u = \rho \, \delta_0 \end{equation}
on the flat torus Eτ with singular strength ρ=8πn.
In Part I (Cambridge J. Math. 3, 2015), a hyperelliptic curve Xˉn(τ)⊂SymnEτ, the Lam\'e curve, associated to the MFE was
constructed. Our construction of Zn(σ;τ) relies on a detailed study
on the correspondence P1←Xˉn(τ)→Eτ induced
from the hyperelliptic projection and the addition map.
As an application of the explicit form of the weight 10 pre-modular form
Z4(σ;τ), a counting formula for Lam\'e equations of degree n=4
with finite monodromy is given in the appendix (by Y.-C. Chou).Comment: 32 pages. Part of content in previous versions is removed and
published separately. One author is remove