Abstract
AbstractWe prove some qualitative results about thep-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 globalp-adic Jacquet–Langlands correspondence can also deal with automorphic forms with principal series representations atpin a nontrivial way, unlike its classical counterpart.
Highlights
Let F be a finite extension of Qp, and let L be a further sufficiently large finite extension of F, which will serve as the field of coefficients
In order to extend Scholze’s construction to admissible unitary Banach space representations Π of GLn(F ), the following seems like a sensible thing to do: choose an open bounded GLn(F )-invariant lattice Θ in Π; Θ/ m is an admissible smooth representation of GLn(F ) on an O-torsion module and we can consider the limit l←im−m Heit(PnC−p 1,FΘ/ m ) equipped with the p-adic topology
Ludwig split at p and ramified at ∞, Scholze shows in [47] that He1t(P1Cp,Fπ) is isomorphic as a GQp × Dp×-representation to the 1st completed cohomology group H1(U p,O/ n) of a tower of Shimura curves associated to a quaternion algebra D, which is ramified at p and split at ∞ and has the same ramification as D0 at all the other places
Summary
Let F be a finite extension of Qp, and let L be a further sufficiently large finite extension of F , which will serve as the field of coefficients. Ludwig split at p and ramified at ∞, Scholze shows in [47] that He1t(P1Cp,Fπ) is isomorphic as a GQp × Dp×-representation to the 1st completed cohomology group H1(U p,O/ n) of a tower of Shimura curves associated to a quaternion algebra D, which is ramified at p and split at ∞ and has the same ramification as D0 at all the other places. If K = Zp, O[[K]] ∼= O[[x]], a commutative formal power series ring in one variable, and the Banach space of continuous functions on K has δ-dimension one and is irreducible in the quotient category, which is equivalent to the category of finite-dimensional vector spaces over the fraction field of O[[x]]; its Schikhof dual is. Isomorphic to O[[x]][1/p], and all irreducible subquotients are finite-dimensional (see, Theorem 1.5)
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