论文标题
分解de la cohomogieétalep-Adique de la tour de de de de de de Drinfeld
Factorization de la cohomologie étale p-adique de la tour de Drinfeld
论文作者
论文摘要
对于有限的扩展名$ f $ $ {\ mathbf q} _p $，德林菲尔德定义了$ {\ mathbb p}^1 \ setMinus {\ mathbb p}^1（f）$的封面塔。对于$ f = {\ mathbf q} _p $，我们描述了该塔的$ p $ - 亚种几何形状的典型同谋，类似于Emerton对模块化曲线塔的完整共同体的分解。至关重要的成分是算术典型的共同体模量P $ P $的有限定理，首先通过对附近周期的计算证明该共同体具有有限的呈现。最后的结果适用于所有$ f $;对于$ f \ neq {\ mathbf q} _p $，这意味着从德林菲尔德塔的共同体获得的$ {\ rm gl} _2（f）$的表示形式与$ f = {\ nathbf q} _p $相反。
For a finite extension $F$ of ${\mathbf Q}_p$, Drinfeld defined a tower of coverings of ${\mathbb P}^1\setminus {\mathbb P}^1(F)$ (the Drinfeld half-plane). For $F = {\mathbf Q}_p$, we describe a decomposition of the $p$-adic geometric étale cohomology of this tower analogous to Emerton's decomposition of completed cohomology of the tower of modular curves. A crucial ingredient is a finitness theorem for the arithmetic étale cohomology modulo $p$ which is shown by first proving, via a computation of nearby cycles, that this cohomology has finite presentation. This last result holds for all $F$; for $F\neq {\mathbf Q}_p$, it implies that the representations of ${\rm GL}_2(F)$ obtained from the cohomology of the Drinfeld tower are not admissible contrary to the case $F = {\mathbf Q}_p$.
