-Indistinguishability on Eigenvarieties

Abstract

In this article, we construct examples of L-indistinguishable overconvergent eigenforms for an inner form of SL2.

On endoscopic p-adic automorphic forms for SL(2)

“…The representation τ ′ from (3) above gives rise to two distinct points x, y ∈ E(K p , m)(Q p ), one of which, say x, is of critical slope whereas the point y is of non-critical slope in the sense of [7,Def.4.4.3]. One verifies this easily using [7, Lemma 4.4.1], a calculation like in Lemma 3.3 of [17] and the fact that…”

“…(1) Π(θ) p is a singleton consisting of a SL 2 (Z p )-unramified representation, (2) for all l ∈ S, Π(θ) l is a supercuspidal L-packet of size two, (3) 2 / ∈ S and (4) Π(θ) ∞ = {D ± k+1 } is a discrete series L-packet of weight k + 1 ≥ 1. We refer the reader to [17,Lemma 4.1] for the construction of examples of such L-packets. Lemma 6.1.…”

“…After possibly shrinking U we may assume that the slope of u p is constant on U . By an argument as in the proof of [17,Theorem 4.3] we may assume that Π(z) is stable for all z ∈ Z, z = x.…”

“…In [17], we constructed non-classical points on certain eigenvarieties for a definite inner form G of SL 2 using endoscopy and a p-adic version of the Labesse-Langlands transfer [16]. We used a local L-packet of size two to change multiplicities in a global L-packet, however in [17] that auxiliary prime was non-archimedean. In this paper we explore the fact that for the split group SL 2 (R) the discrete series L-packets are of size two.…”

“…comes from the definition of the action of H(K p ) ur on H 1 (K p ) m via correspondences. Then Lemma 5.7 implies that we may apply [5,Prop.7.1.1] to the Zariski-dense subset of classical points and the three-dimensional pseudorepresentations that are described in the proof of Lemma 2.2 of [17].…”