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After extending the theory of Rankin-Selberg local factors to pairs of$\ell$-modular representations of Whittaker type, of general linear groups overa non-archimedean local field, we study the reduction modulo $\ell$ of$\ell$-adic local factors and their relation to these $\ell$-modular localfactors. While the $\ell$-modular local $\gamma$-factor we associate to such apair 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 reductionmodulo-$\ell$ of the $\ell$-adic $L$-factor of any Whittaker lifts, but withthe possibility of a strict division occurring. In our main results, wecompletely describe $\ell$-modular $L$-factors in the generic case. We obtaintwo simple to state nice formulae: Let $\pi,\pi'$ be generic $\ell$-modularrepresentations; 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 theinductivity relations for local factors of generic representations. Secondly,we show that \[L(X,\pi,\pi')=\mathbf{GCD}(r_{\ell}(L(X,\tau,\tau'))),\] wherethe divisor is over all integral generic $\ell$-adic representations $\tau$ and$\tau'$ which contain $\pi$ and $\pi'$, respectively, as subquotients afterreduction modulo $\ell$.

Rankin-Selberg local factors modulo $\ell$