Tame Representations of Local Weil Groups and of Chain Groups of Local Principal Orders

Description

We begin by making clear the meaning of the term "tame". The higher ramifi­ cation groups, on the one hand, and the one-units of chain groups, on the other, are to lie in the kernels of the respective representations considered. We shall establish a very natural and very well behaved relationship between representa­ tions of the two groups mentioned in the title, with all the right properties, and in particular functorial under base change and essentially preserving root numbers. All this will be done in full generality for all principal orders. The formal setup for this also throws new light on the nature of Gauss sums and in particular leads to a canonical closed formula for tame Galois Gauss sums. In many ways the "tame" and the "wild" theory have distinct features and distinct points of interest. The "wild" theory is much harder and - as far as it goes at present - technically rather complicated. On the "tame" side, once we have developed a number of new ideas, we get a complete comprehensive theory, from which technical difficulties have disappeared, and which has a naturality, and perhaps elegance, which seems rather rare in this gen,eral area. Among the principal new concepts we are introducing are those of "similarity" of represen­ tations in both contexts and that of the Galois algebra of a principalorder., One might expect that this Galois algebra will ,also be of importance in the wild situation.