Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 18 / 2 / i / 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 $H_t(z,w)=tz\overline{w}$, where $t=H_t(1,1)$ is real. For $\lambda =m+ni$ and $\mu =r+si$ in the period lattice,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 $H_t(z,z)=t|z{|}^2$. The answer isWith 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.