The modular group Γ = PSL(2, ℤ) is the most widely studied of all Fuchsian groups, and several books have been written about Γ, its action on U, and the associated modular functions (meromorphic functions invariant under Γ). The importance of Γ lies in its many connections with other branches of mathematics, and especially with number theory; indeed, interest in Γ first arose out of the investigations of Gauss into numbertheoretic properties of quadratic forms ax2 + bxy + cy2 (a, b, c∈ℤ). We will concentrate on those aspects of Γ which are related to topics considered in earlier chapters; for example, we will apply Γ to the problem (partially dealt with in §4.18) of classifying the compact Riemann surfaces of genus 1.

In §6.1 we construct a bijection between the set of all conformal equivalence classes of tori ℂ/Ω (equivalently, the similarity classes of lattices Ω⊂ℂ) and the orbits of Γ on U. In order to apply this to the Riemann surface S of √p(z) (considered in §4.9), where p is a cubic polynomial, we introduce in §6.2 the discriminant Δ, the non-vanishing of which is equivalent to p having distinct roots, that is, to S having genus 1. We use Δ in §6.3 to obtain an analytic function J:U → ℂ invariant under the action of Γ detailed study in §6.4 of the analytic properties of J enables us to show in §6.5 that J maps U onto ℂ and that the level sets J-1, c∈ℂ, are the orbits of Γ.