Davenport-Hasse relations and an explicit Langlands correspondence

C J Bushnell, G Henniart

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)

Abstract

Let F be a finite extension of Q(P),, and W-F the Well group of F relative to some algebraic closure of Q(p). An irreducible representation of W-F is wildly ramified if its dimension is a power of p and it is not equivalent to an unramified twist of itself. Let g(m)(WT)(F) be the set of equivalence classes of such representations of dimension p(m). Let A(m)(wr)(F) be the set of equivalence classes of irreducible supercuspidal representations pi of GL(p)m(F) such that pi is not a non-trivial unramified twist of itself. The Langlands conjecture predicts the existence of a canonical bijection g(m)(WT)(F) --> A(m)(wr)(F) for each m. Two such bijections are known. The first, denoted L-m,,, is a special case of the construction of a Langlands correspondence due to Harris and Taylor. The other, denoted pi(m),,,, is due to the authors; this construction is quite explicit, and totally different from that of Harris and Taylor. This paper investigates the relation between the two. It is shown that, for sigma is an element of g(m)(wr)(F), the representations pi(sigma), L(sigma) differ by an unramified twist of order dividing p(m). The authors' construction is based on a certain tame lifting map. If this satisfies a certain "Hasse-Davenport relation" on local constants of pairs, then pi(m) = L-m.
Original languageEnglish
Pages (from-to)171 - 199
Number of pages29
JournalJournal fur die Reine und Angewandte Mathematik
Volume519
Publication statusPublished - 2000

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