Local Representations of Galois Groups

Abstract

The purpose of this paper is to present some infinite Galois extensions of algebraic number fields, whose Galois groups possess some matrix representations in local fields, through which the Frobenius automorphisms have algebraic numbers of absolute value N(p)A/2 as characteristic roots. Here p is the prime ideal of the basic field, for which the Frobenius automorphism is considered, and Ix is an integer determined by the representation. These Galois extensions will be generated by the coordinates of the points lying on some Galois coverings of an algebraic variety, which is isomorphic to the quotient of the Siegel upper half space t, by an arithmetic discontinuous group. In the last section of the previous paper [7], we have discussed certain local representations of the Galois groups of such extensions. The present results will give a re-formulation and a refinement of those obtained there.' To be more specific, we consider a certain reductive algebraic group 6 defined over Q which has the following properties: ( 1 ) The center of oQ is isomorphic to the multiplicative group F X of a totally real algebraic number field F. ( 2 ) 63Q acts faithfully on a vector space X over F, where the action of Fx is the same as the center of ok through the isomorphism of (1). ( 3 ) The semi-simple part of 6R is isomorphic to the product of Sp (X1, R) and a compact group. By (3), we can find a subgroup OV of mQ of index 2, which acts on . Let xF denote the ring of algebraic integers in F, and X a finitely generated TFsubmodule of X which spans X over F. Define subgroups F and Fa of W, for every integral ideal a in F, by