Let V be a representation of "the" global Weil group WF, and put
(3.5.1) L{V,s)=\\L(Vvcos),
(V, s) = IT e(Vvcos, /,„ dxv).
The product (3.5.1) converges for s in some right half-plane and
defines a function L(V, s) which is meromorphic in the whole s-plane and satisfies the
functional equation
(3.5.4) L(V, s) = e(V, s)L(V*, 1 - s)
where V* is the dual of V.
For Va quasi-character % this result was proved by Hecke. In the modern version
of his proof ([Tl], [W2]) one shows by Poisson summation that for suitable func-
tions/on A
J y t o a i ! - , X~Hx) dx* =
X(x) dx\
the integrals being defined for all s by analytic continuation. Taking/= \[fv and
using the local functional equation (3.2.1) (with x replaced by %a)s) one finds that
(3.5.4) holds in the "abelian" case, V = x-
At this point, even without having a theory of the local nonabelian
e(VV9 jjv9 dxvY$, one gets, via (2.3.1), (2.3.5), and the inductivity of the local Vs,
that L(Vy s) is meromorphic in the whole plane for each V, being defined by the
product (3.5.1) in a right half-plane, and that L(V, s) is inductive as a function of V.
It follows that
e(V*S} L(V*,l-s)
is inductive in Fand satisfies e'(% s) = Uv e(xvDs- Iv dxv) for quasi-characters x
of A*/F*. It is this fact about the local $(%„, pv, dxv)'s—that their product over all
v for a global x n a s a n inductive extension to all global V—that Deligne uses in
his "global" proof of the existence of local nonabelian e's. Once their existence is
proved, we have e'(V9 s) = s(V, s) by the unicity of inductive functions since e(V9 s),
defined by the product (3.5.2), is inductive in degree 0 by (2.3.5).
(3.5.5) Hecke's global function L(%, s) is entire if x is not of the form cos. Artin
conjectured (in the Galois case) that L{V, s) is entire for any V which has no con-
stituent of the form ws. Weil proved Artin's conjecture for function fields. Recently
Langlands, using ideas of Saito and Shintani, made a first breakthrough in the
number field case, treating certain K's of dimension 2 by base change, using the trace
formula. (See The solution of abase change problem for GL(2) {following Langlands,
Saito, Shintani), these
part 2, pp. 115-133.) These methods work for
all V's of dimension 2 for which the image of WF in PGL( V) is the tetrahedral group.
They also work for some octahedral cases, but a new idea will be needed to apply
them in the nonsolvable icosahedral case. However, J. Buhler [B], with the aid
of the Harvard Science Center PDP11 and the main result of [DS], has proved the
Artin conjecture for one particular icosahedral V of conductor 800, by checking
the existence of the corresponding modular form of weight 1 and level 800.
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