论文标题
在mod $ p $同时组中，$ \ mathrm {gl} _2 $：非避免案例
On the mod $p$ cohomology for $\mathrm{GL}_2$: the non-semisimple case
论文作者
论文摘要
让$ f $是在$ p $以上的所有地方完全不受影响，而$ d $是四元基因代数，它以无限的位置分开或完全是一个无限位置。 Let $\bar{r}:\mathrm{Gal}(\bar{F}/F)\to \mathrm{GL}_2(\bar{\mathbb{F}}_p)$ be a continuous irreducible representation which, when restricted to a fixed place $v|p$, is non-semisimple and sufficiently generic.在一些温和的假设下，我们证明了$ \ mathrm {gl} _2（f_v）$的可接受的平滑表示形式发生在与$ d $相关的Mod $ p $ p $ p $同居的相应的hecke eigenspaces中我们还证明，任何此类表示形式都可以作为$ \ mathrm {gl} _2（f_v）$ - 通过其第一个主要的一致性子组的子空间来代表。如果此外，$ [f_v：\ mathbb {q} _p] = 2 $，我们证明此类表示的长度$ 3 $，证实了Breuil和Paškūnas的猜测。
Let $F$ be a totally real field unramified at all places above $p$ and $D$ be a quaternion algebra which splits at either none, or exactly one, of the infinite places. Let $\bar{r}:\mathrm{Gal}(\bar{F}/F)\to \mathrm{GL}_2(\bar{\mathbb{F}}_p)$ be a continuous irreducible representation which, when restricted to a fixed place $v|p$, is non-semisimple and sufficiently generic. Under some mild assumptions, we prove that the admissible smooth representations of $\mathrm{GL}_2(F_v)$ occurring in the corresponding Hecke eigenspaces of the mod $p$ cohomology of Shimura varieties associated to $D$ have Gelfand-Kirillov dimension $[F_v:\mathbb{Q}_p]$. We also prove that any such representation can be generated as a $\mathrm{GL}_2(F_v)$-representation by its subspace of invariants under the first principal congruence subgroup. If moreover $[F_v:\mathbb{Q}_p]=2$, we prove that such representations have length $3$, confirming a speculation of Breuil and Paškūnas.
