Möbius-Transform of Elliptic Differential

Question

I am currently reading the lecture notes on elliptic functions by Stevenhagen. In exercise 1.7) he considers the elliptic differential $dt/\sqrt{f(t)}$ and one has to show, that there exists a Möbius transformation which brings
\begin{align} f(t)=c t(t-1)(t-\lambda) \end{align} in the form \begin{align} f(t)=(1-t^2)(1-k^2 t^2) \ . \end{align} I can only think of an variable change $t\to 1/\xi^2$ which does that. But I can not show, that this corresponds to a Möbius transformation.
Any help would be appreciated and I would be really glad if you could be explicit since I am no mathematician by any stretch.
Many thanks in advance!

Answer 1

You want a Mobius transformation that sends $0$, $\infty$, $1$ and $\lambda$ to $1$, $-1$, $1/k$ and $-1/k$ in some order (and a $k$ for which this is possible). One can translate this condition on $k$ into a condition about cross-ratios.