Weil group

In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field F, its Weil group is generally denoted WF. There also exists "finite level" modifications of the Galois groups: if E/F is a finite extension, then the relative Weil group of E/F is WE/F = WF/_W (where the superscript c denotes the commutator subgroup). For more details about Weil groups see or or . The Weil group of a class formation with fundamental classes uE/F ∈ H2(E/F, AF) is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program. If E/F is a normal layer, then the (relative) Weil group WE/F of E/F is the extension 1 → AF → WE/F → Gal(E/F) → 1 corresponding (using the interpretation of elements in the second group cohomology as central extensions) to the fundamental class uE/F in H2(Gal(E/F), AF). The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers G/F, for F an open subgroup of G. The reciprocity map of the class formation (G, A) induces an isomorphism from AG to the abelianization of the Weil group. 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. 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). 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).