On some mod $p$ representations of quaternion algebra over $mathbb{Q}_p$
On some mod $p$ representations of quaternion algebra over $mathbb{Q}_p$
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关于 $mathbb{Q}_p$ 上四元数代数的某些 mod $p$ 表示
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发表时间：
2022
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通讯作者：
Haoran Wang
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作者：
Yongquan Hu;Haoran Wang
设F是一个全真实的域，其中p是非分歧的，B是F上至多在一个无穷处分裂的四元数代数.令r：Gal（F/F）→ GL 2（Fp）是满足泰勒-怀尔斯假设的模伽罗瓦表示。假设对于某个固定的地方v|证明了Qp上的四元数代数的容许光滑表示来自与B相关的Shimura簇的mod p上同调，其Gelfand-Kirillov维数为1.作为应用，我们证明了在[Sch 18]中定义的二次Scholze函子在GL 2（Qp）的一般超奇异表示上为零.在可约情形下，我们还证明了Scholze函子象的一些更精细的结构定理。
Let F be a totally real field in which p is unramified and B be a quaternion algebra over F which splits at at most one infinite place. Let r : Gal(F/F ) → GL2(Fp) be a modular Galois representation which satisfies the Taylor-Wiles hypotheses. Assume that for some fixed place v|p, B ramifies at v and Fv is isomorphic to Qp and r is generic at v. We prove that the admissible smooth representations of the quaternion algebra over Qp coming from mod p cohomology of Shimura varieties associated to B have Gelfand-Kirillov dimension 1. As an application we prove that the degree two Scholze’s functor (which is defined in [Sch18]) vanishes on generic supersingular representations of GL2(Qp). We also prove some finer structure theorem about the image of Scholze’s functor in the reducible case.