Rankin–Selberg local factors modulo $$\ell $$ℓ
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.