Abstract
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 with 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. We completely describe ell -modular L-factors in the generic case and 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,π,π′)=L(X,πb,πb′).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} L(X,\\pi ,\\pi ')=L(X,\\pi _b,\\pi _b'). \\end{aligned}$$\\end{document}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′))),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} L(X,\\pi ,\\pi ')=\\mathop {\\mathbf {GCD}}(r_{\\ell }(L(X,\\tau ,\\tau '))), \\end{aligned}$$\\end{document}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 .
Highlights
Let F be a non-Archimedean local field of residual characteristic p and residual cardinality q, and let R be an algebraically closed field of characteristic prime to p or zero
In this article, following Jacquet–Piatetski-Shapiro–Shalika in [9] for complex representations, we associate local Rankin–Selberg integrals with pairs of R-representations of Whittaker type ρ and ρ of GLn(F) and GLm(F) and show that they define L-factors L(X, ρ, ρ ) and satisfy a functional equation defining local γ factors
The purpose of this article lies both in the future study of R-representations by these invariants and in the relationship between -modular local factors and the reductions modulo of -adic local factors
Summary
Let F be a (locally compact) non-Archimedean local field of residual characteristic p and residual cardinality q, and let R be an algebraically closed field of characteristic prime to p or zero. Πis2banχalπ, a1∨nfdorπs2omeχuπn1r∨a,mleitfieedbechthaeraccotmermχoonfrFam×ifi(icnaptiaornindex of π1 and π2 and we have The proof of this theorem is diverse and uses the main result of [13] on test vectors for -adic representations for the banal case, the division inductivity relation of Lfactors for the cuspidal non-supercuspidal case, and a separate examination of the non-banal supercuspidal case by studying their lifts. By restricting to pairs of banal generic representations, we obtain the inductivity relation of L-factors in this setting, and an explicit formula analogous to the -adic case (Theorem 4.17). Our fourth main result gives an equality between the local L-factor of a pair of generic -modular representations we have defined via -modular Rankin–Selberg integrals, and the greatest common divisor of the reductions modulo of certain adic L-factors.
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