Explicit diffeomorphism between complex tori

Question

Let $Im(\tau) > 0$ and $X_{\tau}$ be the complex torus given by $\mathbb{C}/\mathbb{Z}\oplus \tau\mathbb{Z}$. How do I go about constructing an explicit diffeomorphism (as real manifolds) between $X_{\tau}$ and $X_{\tau'}$, where both $\tau$ and $\tau'$ are in the upper half plane? The idea is that such a diffeomorphism in general shouldn't be biholomorphic, since $X_{\tau}$ and $X_{\tau'}$ in general are isomorphic as complex manifolds only if $\tau$ and $\tau'$ are related by an element of $SL(2, \mathbb{Z})$. I constructed the map for some special cases such as $\tau' = \tau+1$ and $\tau' = -1/\tau$, but can't see a way to do it in general. Any hints would be appreciated.

Answer 1

Let $F:\mathbb C\to\mathbb C$ be the $\mathbb R$-linear map such that $F(1)=1$ and $F(\tau)=\tau'$. Explicitly, $$F(z)=z+ \frac{\tau'-\tau}{\tau-\bar \tau}(z-\bar z)$$ The map $F$ induces a map $\tilde F$ from $X_\tau$ to $X_{\tau'}$ because the image of $\mathbb Z\oplus \tau \mathbb Z$ under $F$ is precisely $\mathbb Z\oplus \tau'\mathbb Z$. Also, $F^{-1}$ induces a map from $X_{\tau'}$ to $X_{\tau}$, for the same reason. It is easy to see that the latter map is the inverse of $\tilde F$.