What does an equivalence between tori with $\tau$ and $\tau+1$ mean precisely?

Question

I usually think that complex structure parameters $\tau$ and $\tau+1$ define the same complex torus (as of course do all $\tau'$ related to $\tau$ by modular transformations). I can see that the lattice points defined by two vectors are the same. But doesn't the equivalence between $\tau$ and $\tau+1$ imply that there is some kind of a holomorphic map from the parallelogram drawn on $(\tau,1)$ to a parallelogram drawn on $(\tau+1,1)$? Naively this is not possible, since the angles are clearly not preserved.

Answer 1

No, this equivalence does not mean there is some kind of holomorphic map between the parallelograms.

What it means is there is a holomorphic map between the two quotient tori $$\mathbb C \, / \, (\mathbb Z + \tau \mathbb Z) \quad\text{and}\quad \mathbb C \, / \, (\mathbb Z + (\tau + 1) \mathbb Z) $$ which is completely obvious, given that the lattices $\mathbb Z + \tau \mathbb Z$ and $\mathbb Z + (\tau + 1) \mathbb Z$ are identical.