# Mirror Symmetry of Elliptic Curve

I'm a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason I've been thinking about this, is that I'm doing a computation involving a torus with parameter $$\tau \in \mathbb{H}$$, and my answer in invariant under $$\tau \to \tau+1$$, but not $$\tau \to -1/\tau$$. So I'm thinking that maybe I'm only using the Kähler structure, not the complex structure.

Of course, the complex structure is given by $$\tau \in \mathbb{H}/\rm{PSL}(2, \mathbb{Z})$$. I think for the Kähler structure, we choose a class $$[\omega] \in H^{2}(X,\mathbb{C})$$, which we can parameterize by Kähler parameter

$$t=t_{1} + i \, t_{2},\ t=\frac{1}{2\pi i } \int_{X} [\omega]$$

We identify $$t_{2}>0$$ with the area of the torus. So unlike the complex structure, Kähler structures related by $$\rm{PSL}(2,\mathbb{Z})$$, aren't necessarily identical, correct? After all, one will have small area, the other large.

So I'm confused about how the mirror symmetry acts. If mirror symmetry is some mysterious equivalence of the torus under interchange of the complex and Kähler moduli, doesn't that imply that the space of equivalent Kähler structures is also $$\mathbb{H}/\rm{PSL}(2, \mathbb{Z})$$. Is this correct?

Thank you.