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On some mod p representations of quaternion algebra over ℚp
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 11 / November 2024
- Published online by Cambridge University Press:
- 03 December 2024, pp. 2585-2655
- Print publication:
- November 2024
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- Article
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Let
$F$ be a totally real field in which 
$p$ is unramified and let 
$B$ be a quaternion algebra over 
$F$ which splits at at most one infinite place. Let 
$\overline {r}:\operatorname {{\mathrm {Gal}}}(\overline {F}/F)\rightarrow \mathrm {GL}_2(\overline {\mathbb {F}}_p)$ be a modular Galois representation which satisfies the Taylor–Wiles hypotheses. Assume that for some fixed place 
$v|p$, 
$B$ ramifies at 
$v$ and 
$F_v$ is isomorphic to 
$\mathbb {Q}_p$ and 
$\overline {r}$ is generic at 
$v$. We prove that the admissible smooth representations of the quaternion algebra over 
$\mathbb {Q}_p$ 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 by Scholze [On the 
$p$-adic cohomology of the Lubin–Tate tower, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), 811–863]) vanishes on generic supersingular representations of 
$\mathrm {GL}_2(\mathbb {Q}_p)$. We also prove some finer structure theorems about the image of Scholze's functor in the reducible case.
