论文标题
在某些mod $ p $表示$ \ mathbb {q} _p $上的Quaternion代数的表示
On some mod $p$ representations of quaternion algebra over $\mathbb{Q}_p$
论文作者
论文摘要
让$ f $是一个完全真实的领域，其中$ p $不受影响，而$ b $是四元组代数，最多可以在一个无限的位置拆分。令$ \ edline {r}：\ mathrm {gal}（\ edrolline {f}/f）\ to \ mathrm {gl} _2（\ overline {\ mathbb {f}} _ p）$ be Modular galois表示，使Taylor-wiles-wiles-wiles-wiles hypothessessessage hypotheSesseSessessephessesseptation。假设对于某个固定地点，$ v | p $，$ b $ ramifies in $ v $，$ f_v $是同构至$ \ mathbb {q} _p $和$ \ overline {r} $是$ v $的通用。我们证明，超过$ \ mathbb {q} _p $来自mod $ p $ $ p $ shimura品种的$ \ mathbb {q} _p $的可接受的平滑表示与$ b $相关的gelfand-kirillov dimension $ 1 $。作为一个应用程序，我们证明了两个Scholze的函数在$ \ Mathrm {gl} _2（\ Mathbb {q} _p）$的超词表示上消失。我们还证明了在还原的情况下有关Scholze函数图像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 which splits at at most one infinite place. Let $\overline{r}:\mathrm{Gal}(\overline{F}/F)\to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ 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 vanishes on supersingular representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. We also prove some finer structure theorem about the image of Scholze's functor in the reducible case.
