Classical Modular Curve - Galois Theory of The Modular Curve

Galois Theory of The Modular Curve

The Galois theory of the modular curve was investigated by Erich Hecke. Considered as a polynomial in x with coefficients in Z, the modular equation Φ₀(n) is a polynomial of degree ψ(n) in x, whose roots generate a Galois extension of Q(y). In the case of X₀(p) with p prime, where the characteristic of the field is not p, the Galois group of

Q(x, y)/Q(y)

is PGL₂(p), the projective general linear group of linear fractional transformations of the projective line of the field of p elements, which has p+1 points, the degree of X₀(p).

This extension contains an algebraic extension

of Q. If we extend the field of constants to be F, we now have an extension with Galois group PSL₂(p), the projective special linear group of the field with p elements, which is a finite simple group. By specializing y to a specific field element, we can, outside of a thin set, obtain an infinity of examples of fields with Galois group PSL₂(p) over F, and PGL₂(p) over Q.

When n is not a prime, the Galois groups can be analyzed in terms of the factors of n as a wreath product.