The Picard group of a noncommutative algebraic torus

Let A_q:= C(x^±1 y^±1)/(xy-qyx) . Assuming that q is not a root of unity, we compute the Picard group Pic.Aq/of the algebra Aq, describe its action on the space R.Aq/of isomorphism classes of rank 1 projective modules and classify the algebras Morita equivalent to Aq. Our computations are based on a 'quantum' version of the Calogero-Moser correspondence relating projective Aq-modules to irreducible representations of the double affine Hecke algebras H_t.q-1/2(Sn) at t = 1. We show that, under this correspondence, the action of Pic(A_q) on R(A_q) agrees with the action of SL₂(ℤ) on H_t.q-1/2(Sn) constructed by Cherednik [C1], [C2]. We compare our results with the smooth and analytic cases. In particular, when jqj ≠ 1, we find that Pic(A_q) Auteq(D^b(X))/Z, where D^b(X) is the bounded derived category of coherent sheaves on the elliptic curve X D ℂ/Z.