  We compute Seidel's mirror map for abelian varieties by constructing the
homogeneous coordinate rings from the Fukaya category of the symplectic
mirrors. The computations are feasible as only linear holomorphic disks
contribute to the Fukaya composition in the case of the planar Lagrangians
used. The map depends on a symplectomorphism $\rho$ representing the large
complex structure monodromy. For the example of the two-torus, different
families of elliptic curves are obtained by using different $\rho$ which are
linear in the universal cover. In the case where $\rho$ is merely affine linear
in the universal cover, the commutative elliptic curve mirror is embedded in
noncommutative projective space. The case of Kummer surfaces is also
considered.
