Class Formation - Weil Group

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 u_E/F ∈ H2(E/F, AF) is a kind of modified Galois group, introduced by Weil (1951) and used in various formulations of class field theory, and in particular in the Langlands program.

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 u_E/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.

The reciprocity map of the class formation (G, A) induces an isomorphism from AG to the abelianization of the Weil group.