On the construction of Galois extensions of function fields and number fields

Abstract

This paper consists of two parts and an appendix. In Part 1, we investigate Galois converings and consider the problem of reducing their fields of definition. We restrict ourselves to PSL 2 (Z/pZ)-coverings in Part 2. The results of Part t are applied to obtain Galois extensions with P S L 2 (Z/p Z) as Galois group. We show that if p is an odd prime such that 2, 3 or 7 is a quadratic non-residue modulo p, then PSL2(Z/pZ ) occurs as Galois groups over the rationals. To prove this, Shimura's theory of canonical system of models is used to reduce the fields of definition of certain Galois coverings. Previously, our result is only known for p = 3, 5 and 7. In the appendix, we discuss the classification of Galois coverings, which is necessary in verifying Weil's criterion in certain cases. We also indicate how to use the theory developed in Part t to show Hilbert's result that alternating groups can be realized as Galois groups over Q. This paper is based on the author's doctoral dissertation. He would like to thank Professor Goro Shimura for several valuable suggestion during the course of the research. Notation. For an associative ring S with an identity element, we denote by S x the group of all invertible elements of S.