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.
Read more about this topic: Classical Modular Curve
Famous quotes containing the words theory and/or curve:
“Thus the theory of description matters most.
It is the theory of the word for those
For whom the word is the making of the world,
The buzzing world and lisping firmament.”
—Wallace Stevens (1879–1955)
“I have been photographing our toilet, that glossy enameled receptacle of extraordinary beauty.... Here was every sensuous curve of the “human figure divine” but minus the imperfections. Never did the Greeks reach a more significant consummation to their culture, and it somehow reminded me, in the glory of its chaste convulsions and in its swelling, sweeping, forward movement of finely progressing contours, of the Victory of Samothrace.”
—Edward Weston (1886–1958)