  A pre-modular form $Z_n(\sigma; \tau)$ of weight $\tfrac{1}{2} n(n + 1)$ is
introduced for each $n \in \Bbb N$, where $(\sigma, \tau) \in \Bbb C \times
\Bbb H$, such that for $E_\tau = \Bbb C/(\Bbb Z + \Bbb Z \tau)$, every
non-trivial zero of $Z_n(\sigma; \tau)$, namely $\sigma \not\in E_\tau[2]$,
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 $E_\tau$ with singular strength $\rho = 8\pi n$.
  In Part I (Cambridge J. Math. 3, 2015), a hyperelliptic curve $\bar X_n(\tau)
\subset {\rm Sym}^n E_\tau$, the Lam\'e curve, associated to the MFE was
constructed. Our construction of $Z_n(\sigma; \tau)$ relies on a detailed study
on the correspondence $\Bbb P^1 \leftarrow \bar X_n(\tau) \to E_\tau$ induced
from the hyperelliptic projection and the addition map.
  As an application of the explicit form of the weight 10 pre-modular form
$Z_4(\sigma; \tau)$, a counting formula for Lam\'e equations of degree $n = 4$
with finite monodromy is given in the appendix (by Y.-C. Chou).
