Rankin–Selberg local factors modulo  \(\ell \)

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• 作者：Robert Kurinczuk ; Nadir Matringe
• 关键词：Mathematics Subject ClassificationPrimary 11F70 (Representation ; theoretic methods ; automorphic representations over local and global fields) ; Secondary 22E50 (Representations of Lie and linear algebraic groups over local fields)
• 刊名：Selecta Mathematica
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：23
• 期：1
• 页码：767-811
• 全文大小：816KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1420-9020
• 卷排序：23

文摘

After extending the theory of Rankin–Selberg local factors to pairs of \(\ell \)-modular representations of Whittaker type, of general linear groups over a non-Archimedean local field, we study the reduction modulo \(\ell \) of \(\ell \)-adic local factors and their relation to these \(\ell \)-modular local factors. While the \(\ell \)-modular local \(\gamma \)-factor we associate with such a pair turns out to always coincide with the reduction modulo \(\ell \) of the \(\ell \)-adic \(\gamma \)-factor of any Whittaker lifts of this pair, the local L-factor exhibits a more interesting behaviour, always dividing the reduction modulo-\(\ell \) of the \(\ell \)-adic L-factor of any Whittaker lifts, but with the possibility of a strict division occurring. We completely describe \(\ell \)-modular L-factors in the generic case and obtain two simple-to-state nice formulae: Let \(\pi ,\pi '\) be generic \(\ell \)-modular representations; then, writing \(\pi _b,\pi '_b\) for their banal parts, we have $$\begin{aligned} L(X,\pi ,\pi ')=L(X,\pi _b,\pi _b'). \end{aligned}$$Using this formula, we obtain the inductivity relations for local factors of generic representations. Secondly, we show that $$\begin{aligned} L(X,\pi ,\pi ')=\mathop {\mathbf {GCD}}(r_{\ell }(L(X,\tau ,\tau '))), \end{aligned}$$where the divisor is over all integral generic \(\ell \)-adic representations \(\tau \) and \(\tau '\) which contain \(\pi \) and \(\pi '\), respectively, as subquotients after reduction modulo \(\ell \).Mathematics Subject ClassificationPrimary 11F70 (Representation-theoretic methods; automorphic representations over local and global fields)Secondary 22E50 (Representations of Lie and linear algebraic groups over local fields)1 IntroductionLet F be a (locally compact) non-Archimedean local field of residual characteristic p and residual cardinality q, and let R be an algebraically closed field of characteristic \(\ell \) prime to p or zero. In this article, following Jacquet–Piatetski-Shapiro–Shalika in [9] for complex representations, we associate local Rankin–Selberg integrals with pairs of R-representations of Whittaker type \(\rho \) and \(\rho '\) of \({\text {GL}}_n(F)\) and \({\text {GL}}_m(F)\) and show that they define L-factors \(L(X,\rho ,\rho ')\) and satisfy a functional equation defining local \(\gamma \)-factors. The purpose of this article lies both in the future study of R-representations by these invariants and in the relationship between \(\ell \)-modular local factors and the reductions modulo \(\ell \) of \(\ell \)-adic local factors (we quote our main theorems towards this goal at the end of this introduction).