Weil Group

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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