In this paper, we prove an analogue of Jacquet’s conjecture on the local converse theorem for $$\ell $$ ℓ -adic families of co-Whittaker representations of $${\mathrm {GL}}_n(F)$$ GL n ( F ) , where F is a finite extension of $${\mathbb {Q}}_p$$ Q p and $$\ell \ne p$$ ℓ ≠ p . We also prove an analogue of Jacquet’s conjecture for a descent theorem, which asks for the smallest collection of gamma factors determining the subring of definition of an $$\ell $$ ℓ -adic family. These two theorems are closely related to the local Langlands correspondence in $$\ell $$ ℓ -adic families.

中文翻译：

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关于局部逆定理和族中的下降定理

在本文中，我们证明了 Jacquet 猜想对局部逆定理的一个类比，即 $$\ell $$ ℓ -adic 族的 co-Whittaker 表示的 $${\mathrm {GL}}_n(F)$$ GL n ( F ) ，其中 F 是 $${\mathbb {Q}}_p$$ Q p 和 $$\ell \ne p$$ ℓ ≠ p 的有限扩展。我们还证明了 Jacquet 猜想的一个类比下降定理，它要求确定 $$\ell $$ ℓ -adic 族的定义子环的最小伽马因子集合。这两个定理与 $$\ell $$ ℓ -adic 族中的局部朗兰兹对应关系密切相关。