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:
“Human beings are so made that the ones who do the crushing feel nothing; it is the person crushed who feels what is happening. Unless one has placed oneself on the side of the oppressed, to feel with them, one cannot understand.”
—Simone Weil (1909–1943)
“[The Republicans] offer ... a detailed agenda for national renewal.... [On] reducing illegitimacy ... the state will use ... funds for programs to reduce out-of-wedlock pregnancies, to promote adoption, to establish and operate children’s group homes, to establish and operate residential group homes for unwed mothers, or for any purpose the state deems appropriate. None of the taxpayer funds may be used for abortion services or abortion counseling.”
—Newt Gingrich (b. 1943)