论文标题
在mod $ p $ p $ cohomology上
On the mod $p$ cohomology for $\operatorname{GL}_2$
论文作者
论文摘要
让$ p $为质量数字，而$ f $ a在$ p $以上的地方不受影响的完全真实的字段。令$ \ bar {r}：\ operatorName {gal}（\ bar f/f）\ rightArrow \ rightArow \ propatatorName {gl} _2（\ bar {\ mathbb {f} _p} _p} _p}）$是一个模块化的galois代表，使泰勒 - 韦尔·韦尔斯（Taylor-Wiles）表示满足泰勒 - 韦尔斯（Taylor-Wiles）的假设和某些技术的假设。对于$ v $，$ f $ $ p $上方的固定位置，我们证明，$ \ perperatorname {gl} _2（f_v）$ a $ \ bar {\ mathbb {f_v）$的许多可接受的流畅表示dimension $ [f_v：\ mathbb {q} _p] $。这是基于Breuil-Herzig-hu-Morra-Schraen和Hu-Wang的工作，并扩展了所有情况下的统一证明（$ \ bar {r} $，无论是在$ v $时）。
Let $p$ be a prime number and $F$ a totally real number field unramified at places above $p$. Let $\bar{r}:\operatorname{Gal}(\bar F/F)\rightarrow\operatorname{GL}_2(\bar{\mathbb{F}_p})$ be a modular Galois representation which satisfies the Taylor-Wiles hypothesis and some technical genericity assumptions. For $v$ a fixed place of $F$ above $p$, we prove that many of the admissible smooth representations of $\operatorname{GL}_2(F_v)$ over $\bar{\mathbb{F}_p}$ associated to $\bar{r}$ in the corresponding Hecke-eigenspaces of the mod $p$ cohomology have Gelfand--Kirillov dimension $[F_v:\mathbb{Q}_p]$. This builds on and extends the work of Breuil-Herzig-Hu-Morra-Schraen and Hu-Wang, giving a unified proof in all cases ($\bar{r}$ either semisimple or not at $v$).
