= Solution
Use the convention that a <Hermitian form> is complex linear in its first argument. In the <integral Hermitian forms on the square complex torus> calculation, every <Hermitian form> on $\mathbb C$ has the form $H_t(z,w)=t z\overline w$, where $t=H_t(1,1)$ is real. For $\lambda=m+ni$ and $\mu=r+si$ in the <period lattice>,
$$
E_t(\lambda,\mu)=\operatorname{Im}H_t(\lambda,\mu)=t(nr-ms).
$$
Consequently integrality holds precisely when $t\in\mathbb Z$: necessity follows from $E_t(1,i)=-t$, and sufficiency follows from the displayed formula. Addition of <Hermitian forms> corresponds to addition of $t$. Positivity is equivalent to $t>0$, since $H_t(z,z)=t|z|^2$. \b[The answer is]
$$
\boxed{H_t(z,w)=t z\overline w\quad(t\in\mathbb Z),\qquad H_t>0\iff t>0.}
$$
With the conjugate-linear-first convention the formula is $t\overline z w$ and the sign of $E_t$ reverses. The positive integer parameter is unchanged.
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