Davenport-Hasse relations and an explicit Langlands correspondence, II : twisting conjectures

Abstract

Let F/ℚ p be a finite field extension. The Langlands correspondence gives a canonical bijection between the set 𝒢 F 0 (n) of equivalence classes of irreducible n-dimensional representations of the Weil group 𝒲 F of F and the set 𝒜 F 0 (n) of equivalence classes of irreducible supercuspidal representations of GL n (F). This paper is concerned with the case where n=p m . In earlier work, the authors constructed an explicit bijection π:𝒢 F 0 (p m )→𝒜 F 0 (p m ) using a non-Galois tame base change map. If this tame base change satisfies a certain conjectured automorphic Davenport-Hasse relation, and there exists a Langlands correspondence in p-power degree, then π is the Langlands correspondence. This paper is concerned with the problem of showing, without assuming a priori the existence of the Langlands correspondence, that (on the Davenport-Hasse conjecture) π preserves local constants of pairs, and so is a Langlands correspondence. The principal obstruction is the lack of knowledge of certain elementary properties of the local constant ϵ(π 1 ×π 2 ,s,ψ F ) for π i ∈𝒜 F 0 (p m i ). We state these properties as conjectures (which are certainly true, as consequences of the existence of the Langlands correspondence and analogous properties of the Langlands-Deligne local constant) and show that they imply the desired result: π is a Langlands correspondence. In the process, we prove several new unconditional results concerning π, and give a complete account of the rationality properties of L-functions and local constants of pairs for GL n (F).