Rankin-Selberg local factors modulo $\ell$

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 to 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. In our main results, we completely describe $\ell$-modular $L$-factors in the generic case. We 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 $$L(X,\pi,\pi')=L(X,\pi_b,\pi_b').$$ Using this formula, we obtain the inductivity relations for local factors of generic representations. Secondly, we show that $$L(X,\pi,\pi')=\mathbf{GCD}(r_{\ell}(L(X,\tau,\tau'))),$$ 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 $\ell$.