We construct a certain cross product of two copies of the braided dual (H) over tilde of a quasitriangular Hopf algebra H, which we call the elliptic double E-H, and which we use to construct representations of the punctured elliptic braid group extending the wellknown representations of the planar braid group attached to H. We show that the elliptic double is the universal source of such representations. We recover the representations of the punctured torus braid group obtained in {[}1 5], and hence construct a homomorphism from E-H to the Heisenberg double D-H, which is an isomorphism if H is factorizable. The universal property of EH endows it with an action by algebra automorphisms of the mapping class group <(SL2(Z))over tilde>of the punctured torus. One such automorphism we call the quantum Fourier transform; we show that when H = U-q(g), the quantum Fourier transform degenerates to the classical Fourier transform on D(g) as q --> 1.