On Some Consequences of a Theorem of J. Ludwig

Abstract

We prove some qualitative results about the p-adic Jacquet–Langlands correspondence defined by Scholze, in the $\operatorname {\mathrm {GL}}_2(\mathbb{Q}_p )$ residually reducible case, using a vanishing theorem proved by Judith Ludwig. In particular, we show that in the cases under consideration, the global p-adic Jacquet–Langlands correspondence can also deal with automorphic forms with principal series representations at p in a nontrivial way, unlike its classical counterpart.

Finiteness properties of the category of mod p representations of

“…Since 𝑀/𝜛𝑀 is of finite length, it is enough to show that Hom 𝑘 𝐾 (𝜋 ∨ , 𝑘 𝐾 ) = 0 for every 𝜋 ∈ 𝔅. This follows from [42,Lemma 5.16]. Note that if…”

“…and so 𝜋 𝐾 𝑛 = 𝜋 𝐾 𝑛 as 𝑍 ∩ 𝐾 𝑛 acts trivially on 𝜋; so the argument in [42,Lemma 5.16] carries over to the restriction of 𝜋 to SL 2 (Q 𝑝 ).…”

Gelfand-Kirillov dimension and the $p$-adic Jacquet-Langlands correspondence

“…To ease the notation we will omit the outer brackets, when taking the duals. It is proved in [53,Proposition 6.3] that ( 11) and ( 12) induce a natural isomorphism…”

“…If T ′ acts on S ψ (U p , L/O) and the action extends the action of T univ S , and commutes with the action of (D 0 ⊗ Q p ) × then the argument of Proposition 8.4 shows that S ψ (U p , L/O) m ′ is non-zero. 12 VP would like to thank Yongquan Hu for pointing out that in the statement of [53,Lemma 5.3] and in the last line of its proof χcyc should be replaced by its inverse. 13 In Equation (26) in the proof of [53,Lemma 5.1] λ ∨ should be Hom O (λ, O).…”

“…As in our application, they will actually be injective, we restrict to this simpler setup, in the actual application [55,Corollary 9.3] the representations are not injective, because in the setting of Shimura curves the global units in the centre have to act trivially on the cohomology groups. One way to avoid this issue is to work with unitary groups, but this would be inconvenient for us, as we would like to use the results of [53] proved in the setting of Shimura curves. We fix the issue in Theorem 7.12; again all the fundamental ideas in its proof are due to Scholze.…”