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
Famous quotes containing the words weil and/or group:
“Two prisoners whose cells adjoin communicate with each other by knocking on the wall. The wall is the thing which separates them but is also their means of communication. It is the same with us and God. Every separation is a link.”
—Simone Weil (1909–1943)
“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)