Archimedean local field[edit]

For archimedean local fields the Weil group is easy to describe: for C it is the group C× of non-zero complex numbers, and for R it is a non-split extension of the Galois group of order 2 by the group of non-zero complex numbers, and can be identified with the subgroup C×j C× of the non-zero quaternions.

Finite field[edit]

For finite fields the Weil group is infinite cyclic. A distinguished generator is provided by the Frobenius automorphism. Certain conventions on terminology, such as arithmetic Frobenius, trace back to the fixing here of a generator (as the Frobenius or its inverse).

Local field[edit]

For a local field of characteristic p > 0, the Weil group is the subgroup of the absolute Galois group of elements that act as a power of the Frobenius automorphism on the constant field (the union of all finite subfields).


For p-adic fields the Weil group is a dense subgroup of the absolute Galois group, and consists of all elements whose image in the Galois group of the residue field is an integral power of the Frobenius automorphism.


More specifically, in these cases, the Weil group does not have the subspace topology, but rather a finer topology. This topology is defined by giving the inertia subgroup its subspace topology and imposing that it be an open subgroup of the Weil group. (The resulting topology is "locally profinite".)

Function field[edit]

For global fields of characteristic p>0 (function fields), the Weil group is the subgroup of the absolute Galois group of elements that act as a power of the Frobenius automorphism on the constant field (the union of all finite subfields).

Number field[edit]

For number fields there is no known "natural" construction of the Weil group without using cocycles to construct the extension. The map from the Weil group to the Galois group is surjective, and its kernel is the connected component of the identity of the Weil group, which is quite complicated.

Langlands group

Shafarevich–Weil theorem

; Tate, John (2009) [1952], Class field theory, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4426-7, MR 0223335

Artin, Emil

(1973), "Les constantes des équations fonctionnelles des fonctions L", Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture notes in mathematics, vol. 349, Berlin, New York: Springer-Verlag, pp. 501–597, doi:10.1007/978-3-540-37855-6_7, ISBN 978-3-540-06558-6, MR 0349635

Deligne, Pierre

Kottwitz, Robert (1984), "Stable trace formula: cuspidal tempered terms", Duke Mathematical Journal, 51 (3): 611–650,  10.1.1.463.719, doi:10.1215/S0012-7094-84-05129-9, MR 0757954

CiteSeerX

Rohrlich, David (1994), "Elliptic curves and the Weil–Deligne group", in Kisilevsky, Hershey; Murty, M. Ram (eds.), Elliptic curves and related topics, CRM Proceedings and Lecture Notes, vol. 4, , ISBN 978-0-8218-6994-9

American Mathematical Society

Tate, J. (1979), , Automorphic forms, representations, and L-functions Part 2, Proc. Sympos. Pure Math., vol. XXXIII, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 978-0-8218-1435-2

"Number theoretic background"

(1951), "Sur la theorie du corps de classes (On class field theory)", Journal of the Mathematical Society of Japan, 3: 1–35, doi:10.2969/jmsj/00310001, ISSN 0025-5645, reprinted in volume I of his collected papers, ISBN 0-387-90330-5

Weil, André