Lets $E\_{\tau}^{\rho}$ be the elliptic curve with complex structure given by $\tau$ in upper half plane and complexified Kahler form $\rho \frac{dz\wedge d\bar{z}}{2}$.( $\rho$ is in upper half plane too)


Then mirror symmetry says that mirror to $E\_{i}^{\rho}$ in A-side is $E^{i}\_{\rho}$ in B-side.(see the paper of Polishchuk and Zaslow)


then what is the mirror for general $E\_{\tau}^{\rho}$ in A-side (i.e. when we change the complex structure on A-side from the one given by $i$ to something else)??



Mirror symmetry for elliptic curves
The mirror of $E^\rho\_\tau$ is $E^\tau\_\rho$, as you may have guessed. The reason this is not discussed in, say, Polishchuk/Zaslow is that the derived category does not depend on the symplectic structure, and the Fukaya category does not depend on the complex structure, so for their purposes the parameter $\tau$ is irrelevant.


Btw, another article on "Mirror symmetry and elliptic curves" that you might find interesting is due to Dijkgraaf's, with title as indicated; it is contained in a 1995 volume on "The moduli space of curves" (see e.g. <http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.8.4194&rep=rep1&type=pdf>)


