Arithmetic of 3 and 4 Branch Point Covers

Abstract

The method of choice nowadays for achieving a group G as a Galois group of a regular extension of Q(x) goes under the heading of rigidity. It works essentially, only, to produce Galois extensions of Q(x) ramified over 3 points. The three rigidity conditions ((0.1) below) imply that G is generated in a very special way by two elements. Generalization of rigidity that considers extensions with any number r of branch points has been around even longer than rigidity (§5.1). Of the three conditions, the generalization of the transitivity condition, 0.1 c), requires only the addition of an action of the Hurwitz monodromy group H _r (a quotient of the Artin braid group). But it also adds a 4th condition that in many situations amounts to asking for a Q-point on the Hurwitz space associated the data for the generators of G. Theorem 1 below -our main theorem- is that in the case r = 4 this is equivalent to finding a Q-point on a curve derived from a quotient of the upper half plane by a subgroup of PSL₂(ℤ).