论文标题
通信thêtalocale $ \ ell $ -Modulaire i：GroupeMétaplectique，重新定义de weil et $θ$ -LIFT
Correspondance thêta locale $\ell$-modulaire I : groupe métaplectique, représentation de Weil et $Θ$-lift
论文作者
论文摘要
让$ f $是一个剩余的型型型型型型$ p $的本地非阿基米人或有限的领域，但其特征与$ 2 $不同。让$ w $成为$ f $以上有限尺寸的符合空间。假设$ r $是特征性$ \ ell \ neq p $的字段，因此存在非微不足道的平滑添加字符$ψ：f \ to r^\ times $。然后，海森伯格集团$ h（W）$的石像Neumann定理对于具有$ r $ $ r $系数的表示形式仍然有效。它导致$ \ text {sp}（w）$的投影表示形式，该$将$ \ text {sp}（w）$ by $ r^\ times $的中央扩展名的真正平滑表示：这是Metapclectic组的模块化Weil表示。对于任何双对$（H_1，H_2）$，它们向Metapclect组的升降机可能不会根据危险的不同案例旋转。最终，计算模块化Weil表示的最大同种型商允许定义$θ$ -LIFT。因此，这些新工具可以提供一些新的调查线，例如研究标量扩展和减少模型$ \ ell $。
Let $F$ be a field which is, either local non archimedean, or finite, of residual charcateristic $p$ but of characteristic different from $2$. Let $W$ be a symplectic space of finite dimension over $F$. Suppose $R$ is a field of characteristic $\ell \neq p$ so that there exists a non trivial smooth additive character $ψ: F \to R^\times$. Then the Stone-von Neumann theorem of the Heisenberg group $H(W)$ is still valid for representations with coefficients in $R$. It leads to a projective representation of the group $\text{Sp}(W)$ which lifts to a genuine smooth representation of a central extension of $\text{Sp}(W)$ by $R^\times$: this is the modular Weil representation of the metaplectic group. For any dual pair $(H_1,H_2)$, their lifts to the metaplectic group may splitor not according to the different cases at stake. Eventually, computing the biggest isotypic quotient of the modular Weil representation allows to define the $Θ$-lift. Some new lines of investigation are thus available with these new tools such as studying scalar extension and reduction modulo $\ell$.
