A p-adic Labesse–Langlands transfer

Abstract

Abstract. We prove a p-adic Labesse-Langlands transfer from the group of units in a definite quaternion algebra to its subgroup of norm one elements. More precisely, given an eigenvariety for the first group, we show that there exists an eigenvariety for the second group and a morphism between them that extends the classical Langlands transfer. In order to find a suitable target eigenvariety for the transfer we formalise a notion of Langlands compatibility of tame levels. Proving the existence of Langlands compatible tame levels is the key to pass from the classical transfer on the level of L-packets to a map between classical points of eigenvarieties, which is then amenable for interpolation to give the p-adic transfer.

-Indistinguishability on Eigenvarieties

“…Part (2) can be proved in the same way as Proposition 4.16 of [10]. Namely for an automorphic representation π of G(A) with e · π p f = 0 and any τ ∈ Π( π p ) define…”

“…There are natural Q-structures on the Hecke algebras defined by the subalgebras of Q-valued functions which we denote by H Q,S , H Q,ur,S etc. The monomorphism λ can in fact be defined over Q (see Section 2 of [10] for details). In particular we constructed in Lemma 2.10 of [10] an inclusion of Q-algebras…”

“…The monomorphism λ can in fact be defined over Q (see Section 2 of [10] for details). In particular we constructed in Lemma 2.10 of [10] an inclusion of Q-algebras…”

“…This is proved as usual (cf. [5,Section 6.4.5] and also [10,Prop. 3.9] for a proof of the analogous assertion for G in the same notation) using the fact that forms of small slope are classical and that classical weights are dense in weight space.…”

“…Proof. Part (1) follows from Proposition 4.15 and Proposition 4.16 of [10], once we remark that for any (3) follows from [10] Theorem 5.7 and the proof of it.…”