Proceedings of Symposia in Pure Mathematics
Vol. 33 (1979), part 2, pp. 3-26
NUMBER THEORETIC BACKGROUND
J. TATE
1. Weil Groups. If G is a topological group we shall let
Gc
denote the closure of
the commutator subgroup of G, and Gah = G/Gc the maximal abelian Hausdorff
quotient of G. Recall that if H is a closed subgroup of finite index in G there is a
transfer homomorphism t: Gah - i/ab, defined as follows: if s: H\G - G is any
section, then for ge G,
t(g&) = II hgtS (mod H*\
x^H\G
where hgtX e H is defined by s(x)g = hgtXs (xg).
(1.1) Definition of Weil group. Let Fbe a local or global field and F a separable
algebraic closure of F. Let E, £",••• denote finite extensions of Fin F. For each such
E, let GE = Ga\(F/E). A Weil group for F/F is not really just a group but a triple
(WF, p, {/*£•}). The first two ingredients are a topological group WF and a continu-
ous homomorphism p\ WF -• GF with dense image. Given WF and p, we put
WE =
p~l(GE)
for each finite extension E of F in F. The continuity of ip just means
that WE is open in WF for each is, and its having dense image means that p induces
a bijection of homogeneous spaces:
WF/WE _^ _ GF/GE * HomF(F, F)
for each is, and in particular, a group isomorphism WF/WE « Ga\(E/F) when £"/F
is Galois. The last ingredient of a Weil group is, for each E, an isomorphism of
topological groups rE:CE^ Wf, where
c
_ (The multiplicative group E* of is in the local case,
Ithe idele-class group A%/E* in the global case.
In order to constitute a Weil group these ingredients must satisfy four conditions:
(Wj) For each E, the composed map
CE JL Wt
iBdttCedbyff,
Gf
is the reciprocity law homomorphism of class field theory.
(W2) Let w e WF and a = p(w) e GF. For each E the following diagram is com-
mutative :
AMS (MOS) subject classifications (1970). Primary 12A70.
© 1979, American Mathematical Society
3
http://dx.doi.org/10.1090/pspum/033.2/546607
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