2014/08/22 by Robert Kurinczuk, Kurinczuk, Robert, Nadir Matringe +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1408.5252
openalex publication_date 2014/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,πb'). 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.