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.
Read more about Weil Group: Weil Group of An Archimedean Local Field, Weil Group of A Finite Field, Weil Group of A Local Field, Weil Group of A Function Field, Weil Group of A Number Field, Weil–Deligne Group, Langlands Group, See Also
Other articles related to "group, weil group, weil, weil groups, groups":
... More precisely the Artin map gives an isomorphism from the group GL1(K)= K* to the abelianization of the Weil group ... smooth representations of GL1(K) are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL1(C) ... This gives the Langlands correspondence between homomorphisms of the Weil group to GL1(C) and irreducible smooth representations of GL1(K) ...
... This is not a Weyl group and has no connection with the Weil-Châtelet group or the Mordell-Weil group The Weil group of a class formation with fundamental classes uE/F ∈ H2(E/F ... If E/F is a normal layer, then the Weil group U of E/F is the extension 1 → AF → U → E/F → 1 corresponding to the fundamental class uE/F in H2(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 ...
... If K is a local or global field, the theory of class formations attaches to K its Weil group WK, a continuous group homomorphism φ WK → GK, and an isomorphism of topological groups where CK is K× or ... of residue characteristic p ≠ ℓ, then it is simpler to study the so-called Weil–Deligne representations of WK ...
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