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We construct explicitly the quantization of classical linear maps of $SL(2,R)$ on toroidal phase space, of arbitrary modulus, using the holomorphic(chiral) version of the metaplectic representation. We show that Finite QuantumMechanics (FQM) on tori of arbitrary integer discretization, is a consistentrestriction of the holomorphic quantization of $SL(2, Z)$ to the subgroup$SL(2, Z)/\Gamma_l$, $\Gamma_l$ being the principal congruent subgroup mod l,on a finite dimensional Hilbert space. The generators of the ``rotation group''mod l, $O_{l}(2)\subset SL(2,l)$, for arbitrary values of l are determined aswell as their quantum mechanical eigenvalues and eigenstates.Comment: 12 pages LaTeX (needs amssymb.sty). Version as will appear in J.  Phys.

Holomorphic quantization on the torus and finite quantum mechanics