For i ramified,
e(x. /
= f Z"'(*) / (*) dx
2 f x~i(x) $ (x) dx
= f , x~Kx)l(x)dx.
From these formulas one deduces, for % arbitrary and co unramified
( e(xt), I, dx,) = e(x, ]), dx) a){iz»W+«W).
(3.3) Local nonabelian L-functions. We owe to Artin the discovery that there is
an inductive (2.3.2) function L of representations of Weil groups of local fields
such that L(V)= L{y) when Fis a representation of degree 1 corresponding to the
quasi-character ^. The explicit description of L is as follows:
(3 3.1) F archimedean. Since L is additive, we can define it by giving its value on
irreducible V. For F complex, WF = F* is abelian, and the only irreducible K's are
the quasi-characters %, for which L has already been defined. For Freal, the only
irreducible K's which are not of dimension 1 are those of the form V Ind^/F #,
where % is a quasi-character of F* = WF which is not invariant under "complex
conjugation". For such V we put L(V) = L(%), as we are forced to do in order that
L be inductive.
(3.3.2) F nonarchimedean. Let / be the inertia subgroup of WF. Let 0 be an
"inverse Frobenius", i.e., an element of WF such that ||0|| = ||^r||F. This condition
determines 0 uniquely mod J and we put L(V) = det(l 0\ F7)-1, where V1 is the
subspace of elements in V fixed by /.
A proof that the "nonabelian" function L defined as above is inductive can be
found in [D3, Proposition 3.8] (as well as in [A]). In the archimedean case one uses
the relation r^s) = rR(s)rR(s 4- 1). Technically, in order that L have values in a
group, we should view L as a function which associates with V the meromorphic
functions *-* L(Va)s), and take the X in Definition (2.3.2) to be the multiplicative
group of nonzero meromorphic functions of s.
(3.4) The local "nonabelian" e-function, e(V, ]), dx). For this there is at present
only an existence theorem (see below), no explicit formula.1 This lack is not sur-
prising if we recall that the formulas defining e in (3.2) make essential use of the
interpretation of % as a quasi-character of F*; if we think of % as a quasi-character
of WF we have no way to define e(%, cj, dx) without using the reciprocity law iso-
morphism F* « Wp. In fact it was his idea about "nonabelian reciprocity laws"
relating representations of degree n of WF to irreducible representations it of
GL(«, F), and the possibility of defining e{%, (J), dx) for the latter, which led Lang-
lands to conjecture and prove a version of the following big
There is a unique function e which associates with each choice of
a local field F, a nontrivial additive character (jj of F, an additive Haar measure dx
on Fanda representation Vof WF a number e{V, cjj, dx) e C* such that e(V, (J), dx)
e{%, (j), dx) if V is a representation of degree 1 corresponding to a quasi-character %,
and such that if F is a local field and we choose for each finite separable extension E
Except for Deligne's expression in terms of Stiefel-Whitney classes for orthogonal representa-
tions [D5, Proposition 5.2].
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