Inverse Galois Problem - A Construction With An Elliptic Modular Function

A Construction With An Elliptic Modular Function

Let n be any integer greater than 1. A lattice Λ in the complex plane with period ratio τ has a sublattice Λ' with period ratio nτ. The latter lattice is one of a finite set of sublattices permuted by the modular group PSL(2,Z), which is based on changes of basis for Λ. Let j denote the elliptic modular function of Klein. Define the polynomial φ_n as the product of the differences (X-j(Λ_i)) over the conjugate sublattices. As a polynomial in X, φ_n has coefficients that are polynomials over Q in j(τ).

On the conjugate lattices, the modular group acts as PGL(2,Z_n). It follows that φ_n has Galois group isomorphic to PGL(2,Z_n) over Q(J(τ)).

Use of Hilbert's irreducibility theorem gives an infinite (and dense) set of rational numbers specializing φ_n to polynomials with Galois group PGL(2,Z_n) over Q. The groups PGL(2,Z_n) include infinitely many non-solvable groups.