interpretation of Hecke's functional equation using ideles established Hasse's con-
jecture for abelian L-functions. For non-abelian L-functions, the factorization of
the root number into a product of local root numbers was achieved by Dwork up to
sign, and in full generality by Langlands. Greatly simplified analytic proofs were
subsequently produced by Deligne and by Tate.
Langlands' original approach, itself a lengthy and intricate induction which
builds on earlier work of Dwork as well as on his own new Gauss sum identities
is noteworthy because in it the existence of local root numbers is established by
purely local techniques, suggestive of the existence of a non-abelian local class field
theory, a theme which is at the heart of Langlands' Functoriality Principle.
The Appendix surveys in broad outline the two main components of the local
theory of root numbers: (a) a description of the form of the three main local root
number identities, (b) the determination of the corresponding three generators for
the kernel of Brauer induction for solvable groups. The main goal of this Appendix
is to introduce the reader to one of the most interesting and profound tools for the
study of L-functions and automorphic forms.
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