The modular group of first level Γ1 = PSL(2, ℤ) made its debut in Section 2.6 in connection with the conformal equivalence of tori: Two such tori are equivalent if and only if their period ratios are related by a substitution of this kind. The quotient of the upper half-plane ℍ by the action of _1, and by the further modular groups to be introduced later in this section, produces a whole series of interrelated curves whose function fields (modular functions) have deep arithmetic and geometric applications going back to Abel [1827] and Jacobi [1829]. The two most striking of these are Hermite's solution of the general polynomial equation of degree 5 [1859], explained in Chapter 5, and Weber's realization [1891] of Kronecker's youthful dream (1860) of describing the absolute class field of the imaginary quadratic field for a positive square-free integer D; see Chapter 6. The present section prepares the way. Fricke and Klein [1926] and Shimura [1971] are recommended for information at a more advanced level.

The Modular Group of First Level

A modular function of first level is a function f of rational character in the open upper half-plane ℍ that is invariant under the action of Γ1 = PSL(2, ℤ), that is, f ((aω + b)/(cω + d)) = f(ω) for every ω ∈ ℍ and [ab/cd] ∈ Γ1; such a function is of period 1 because [11/01]: ω → ω + 1 is a modular substitution, so it is possible to expand it in powers of in the vicinity of, as in.