We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalization group flow, we derive many properties of both the integer and fractional quantum Hall effects including: universality; the selection rule |p₁q₂-p₂q₁| = 1 for transitions between quantum Hall states characterized by filling factors ₁ = p₁/q₁ and ₂ = p₂/q₂; critical values of the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalization group flow lead to the semicircle rule for transitions between Hall plateaux.