Rankin–Selberg local factors modulo ℓ

After extending the theory of Rankin–Selberg local factors to pairs of ℓ-modular representations of Whittaker type, of general linear groups over a non-archimedean local field, we study the reduction modulo ℓ of ℓ-adic local factors and their relation to these ℓ-modular local factors. While the ℓ-modular local γ-factor we associate to such a pair turns out to always coincide with the reduction modulo ℓ of the ℓ-adic γ-factor of any Whittaker lifts of this pair, the local L-factor exhibits a more interesting behaviour; always dividing the reduction modulo-ℓ of the ℓ-adic L-factor of any Whittaker lifts, but with the possibility of a strict division occurring. In our main results, we completely describe ℓ-modular L-factors in the generic case. We obtain two simple to state nice formulae: Let π, π′ be generic ℓ-modular representations; then, writing πb, π′ b for their banal parts, we have L(X, π, π′ ) = L(X, πb, π′ b ).

Using this formula, we obtain the inductivity relations for local factors of generic representations. Secondly, we show that L(X, π, π′ ) = GCD(rℓ(L(X, τ, τ′
))), where the divisor is over all integral generic ℓ-adic representations τ and τ
′ which contain π and π ′ , respectively, as subquotients after reduction modulo ℓ.