Let $L$ be a Lattice in the complex plane $\mathbb{C}$ . Let $L=\langle1,\tau\rangle$, where $\tau\in H $( Upper half plane). Given that $iL=L$. What are the possible values of $\tau$?
So , by hypothesis $i\in L$ then $i=m+n\tau\implies \tau =\frac{i-m}{n}$ I think I have to proceed like this.
Note that you also need $-i\tau\in L$, and $-i\tau=\frac{1+im}{n} = m\tau + \frac{m^2+1}{n}$, so $\tau=\frac{i-m}{n}$ not just for any integer $m$ and positive integer $n$, but only if $n|(m^2+1)$.
You can also say that such lattice has to be square, so $\tau$ has to be equal to some Möbius transformation applied to $i$: $\tau=\frac{ai+b}{ci+d}$ for integer $a,b,c,d$ such that $ad-bc=1$ (this representation is not unique: there are $4$ ways to choose $a,b,c,d$ which give any particular $\tau$); equivalently, $\tau = \frac{i+ac+bd}{c^2+d^2}$. This answer is equivalent to the answer given by the first approach. In particular, you can verify that the denominator $c^2+d^2$ indeed divides $(ac+bd)^2+1$, taking into account $1=(ad-bc)^2$.