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Local fields of characteristic \(p\) as limits of local fields of characteristic 0. (Les corps locaux de caractéristique \(p\), limites de corps locaux de caractéristique 0.) (French) Zbl 0578.12014

Représentations des groupes réductifs sur un corps local, Travaux en Cours, 120-157 (1984).
[For the entire collection see Zbl 0544.00007.]
D. Kazhdan has suggested that the representation theory of a reductive group over a local field of positive characteristic should be obtained as the ”limit” of the representation theory of groups over fields with characteristic zero and the same residue field. The limit involves letting the absolute ramification index tend to infinity. The present paper proves the results which should correspond to this, via functoriality, for Galois representations, establishing appropriate machinery in the process.
M. Krasner introduced a version of this “limiting” idea [C. R. Acad. Sci., Paris 224, 173–175, 434–436 (1947; Zbl 0038.17701, Zbl 0038.17702)], but the present approach is better suited to the question at hand and avoids the technical unpleasantness of multivalued arithmetic.
The paper proves the equivalence of certain categories, and shows that the equivalence satisfies various compatibilities (\(L\)- and \(\varepsilon\)- factors, for example...).
An appendix extends the standard theory of ramification groups to the case of non-Galois extensions.
Reviewer: J.Repka

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
11S15 Ramification and extension theory
11F85 \(p\)-adic theory, local fields