Weil Group

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The Weil group of a class formation with fundamental classes uE/FH2(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 → AFWE/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

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... 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 ...
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... 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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