Rankin–Selberg local factors modulo $$\ell $$ℓ
Rankin–Selberg local factors modulo $$\ell $$ℓ
复制标题
Rankin-Selberg 局部因子模 $$ell $$ℓ
DOI：
10.1007/s00029-016-0258-6
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
N. Matringe
中科院分区：
文献类型：
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作者：
R. Kurinczuk;N. Matringe
将Rankin-Selberg局部因子理论推广到$$\ell $$研究了非阿基米德局部域上一般线性群的惠特克型的l -模表示$$\ell $$的$$\ell $$输入局部因子及其与它们的关系$$\ell $$l -模局部因子。而$$\ell $$局部模$$\gamma $$γ因子，我们与这样的一对相关联，结果总是与还原模一致$$\ell $$的$$\ell $$进位$$\gamma $$这对的任何Whittaker提振的γ-因子，局部l因子表现出更有趣的行为，总是除约化模-$$\ell $$的$$\ell $$任意惠特克提升的，但有严格除法发生的可能性。我们完全描述了$$\ell $$在一般情况下的模l因子，得到两个简单的状态好的公式$$\pi ,\pi '$$π '是一般的$$\ell $$-模表示；然后，写作$$\pi _b,\pi '_b$$πb和πb '表示它们的平凡部分，我们有 $$\begin{aligned} L(X,\pi ,\pi ')=L(X,\pi _b,\pi _b'). \end{aligned}$$L(X,π,π ')=L（X,πb,πb '）利用这个公式，我们得到了一般表示的局部因子的归纳关系。其次，我们展示了 $$\begin{aligned} L(X,\pi ,\pi ')=\mathop {\mathbf {GCD}}(r_{\ell }(L(X,\tau ,\tau '))), \end{aligned}$$L(X,π,π ')=GCD(r r (L(X,τ,τ '))，其中除数在所有积分泛型上$$\ell $$进位表示$$\tau $$τ和$$\tau '$$τ '包含$$\pi $$π和$$\pi '$$，分别为约简模后的子商$$\ell $$r。
After extending the theory of Rankin–Selberg local factors to pairs of $$\ell $$ℓ-modular representations of Whittaker type, of general linear groups over a non-Archimedean local field, we study the reduction modulo $$\ell $$ℓ of $$\ell $$ℓ-adic local factors and their relation to these $$\ell $$ℓ-modular local factors. While the $$\ell $$ℓ-modular local $$\gamma $$γ-factor we associate with such a pair turns out to always coincide with the reduction modulo $$\ell $$ℓ of the $$\ell $$ℓ-adic $$\gamma $$γ-factor of any Whittaker lifts of this pair, the local L-factor exhibits a more interesting behaviour, always dividing the reduction modulo-$$\ell $$ℓ of the $$\ell $$ℓ-adic L-factor of any Whittaker lifts, but with the possibility of a strict division occurring. We completely describe $$\ell $$ℓ-modular L-factors in the generic case and obtain two simple-to-state nice formulae: Let $$\pi ,\pi '$$π,π′ be generic $$\ell $$ℓ-modular representations; then, writing $$\pi _b,\pi '_b$$πb,πb′ for their banal parts, we have $$\begin{aligned} L(X,\pi ,\pi ')=L(X,\pi _b,\pi _b'). \end{aligned}$$L(X,π,π′)=L(X,πb,πb′).Using this formula, we obtain the inductivity relations for local factors of generic representations. Secondly, we show that $$\begin{aligned} L(X,\pi ,\pi ')=\mathop {\mathbf {GCD}}(r_{\ell }(L(X,\tau ,\tau '))), \end{aligned}$$L(X,π,π′)=GCD(rℓ(L(X,τ,τ′))),where the divisor is over all integral generic $$\ell $$ℓ-adic representations $$\tau $$τ and $$\tau '$$τ′ which contain $$\pi $$π and $$\pi '$$π′, respectively, as subquotients after reduction modulo $$\ell $$ℓ.