Galois e-factors

119

We normalize the reciprocity map t^.W^-^ E"" of local class field theory so that geometric Frobenius elements of W^ correspond to uniformizers of E. If cr is a representation of W^, we let г {о) be the 8-factor of о for the additive character 11/^ = 11/0 Tr^if of E (whose order we write v^) and the Haar measure on E self-dual for if/E, it is a monomial in q^^ and we let е'((т) be its value at ^ = 1/2. For a unitary, 8'(<j) is of absolute value 1. One writes a (a) for the exponent of the Artin conductor of a, Det a for the character of E"^ which verifies (Det a) о i^ = det о a. For every character x of E"", we let /ö" be the representation g^ x° ^Eis)(^ig) of We ; one says that a is minimal if one has a (x(j) ^ a (x) foi" every character xofE""; the minimal exponent a^^^(a) of cr is the minimum of а{х(т) when x runs through characters of E"".

We let IjE denote the trivial representation of W^ or the trivial character of E"". If Ä^ is a finite extension of E in F, we consider W^^ as a subgroup of W^. We denote by Indf the induction of representations from Wj^ to W^, and by a suffix К the restriction to PF^^ of representations of W^. One puts öf;^jE = Det (Indf 1^^) and Я|^/£ = 8'(Indf Ijj^); also d^^jE is the differential exponent of К over E, Cj^ie the ramification index.

When E is F, we shall feel free to drop the index F.

2 . A Gauss sum

Let £" be a finite extension of F in F.

Let xeE"". Suppose first;? Ф 2 and Ve {x) + v^ odd. Let b be an integer such that ^£ W + ^£ + 2Й + 1 =0, and CD a uniformizer of E. Then one has ц/е {хсЬ ^^(^) = 1 for ^ePe and one can consider the sum ф(х) = I^\j/e{x(X)^^^^I2), taken over a system of representatives in Re of Re/Pe- In fact, (p(x) is clearly a "Gauss sum" over kE [GK, p. 4]. From the classical properties of Gauss sums over finite fields, it is apparent that (p(x) depends only on the class of x in C£®Z/2Z, and

( P ( X ) 2= ( - 1 ) <«H - 1 ) / ^^^ .

One can hence define a function Ge on E, factoring through Q ® Z/2Z in the following way:

If / 7 is odd, Ge(x) = 1 if t;^(x) -f v^ is even, and Ge{x) = qÊ^'^ (p (x), with cp (x) as above, if t;^ (x) + v^ is odd. Ifp = 2, Ge (x) = 1 for all x. In each case, Ge (x)"^ = 1. One also writes Ge for the function on Q®Z[l/p] deduced from Ge- The valuafion Ve defines a map Q ® Z [1/p] -> Z [1//?], still denoted by Ve .

3 . Study of Q

Let Ä^ be a finite extension of E in F. One has a canonical injection of Q into C|^ induced by the inclusion of E in K.

Lemma 1. Suppose К is a finite Galois extension of E, with group G. Then the injection of Ce into Q induces an isomorphism of Се®Ъ [1//?] with the set of fixed points {C^ ® Z [1/p])^ of G onC^®Z [1/p]. In particular one gets an injection ofC^ into Ce®Z[1/p].