Rankin–Selberg local factors modulo ℓ

After extending the theory of Rankin–Selberg local factors to pairs of -modular representations of Whittaker type, of general linear groups over a non-Archimedean local field, we study the reduction modulo of -adic local factors and their relation to these -modular local factors. While the -modular local γ -factor we associate with such a pair turns out to always coincide with the reduction modulo of the -adic γ -factor of any Whittaker lifts of this pair, the local L-factor exhibits a more interesting behaviour, always dividing the reduction modulo- of the -adic L-factor of any Whittaker lifts, but with the possibility of a strict division occurring. We completely describe -modular L-factors in the generic case and obtain two simple-to-state nice formulae: Let π ,π ' be generic -modular representations; then, writing π b,π 'b for their banal parts, we have aligned L(X,π ,π ')=L(X,π b,π b'). aligned Using this formula, we obtain the inductivity relations for local factors of generic representations. Secondly, we show that aligned L(X,π ,π ')= GCD(r(L(X,τ ,τ '))), aligned where the divisor is over all integral generic -adic representations τ and τ ' which contain π and π ' , respectively, as subquotients after reduction modulo .