We establish the stability of Rankin–Selberg local $\gamma $-factors attached to generic representations of symplectic groups and general linear groups over $p$-adic fields. Our approach uses the Langlands–Shahidi method and provides a new proof that does not rely on previously known stability results. The key innovations involve handling the geometry of orbit spaces with non-trivial stabilizers and developing new techniques for analyzing partial Bessel integrals in this setting. The results have important implications for the Langlands program, particularly for local converse theorems and the functoriality conjecture.
Nous établissons la stabilité des facteurs $\gamma $ locaux de Rankin–Selberg attachés aux représentations génériques des groupes symplectiques et des groupes généraux linéaires sur les corps $p$-adiques. Notre approche utilise la méthode de Langlands–Shahidi et fournit une nouvelle preuve qui ne repose pas sur les résultats de stabilité précédemment connus. Les innovations clés incluent la maîtrise de la géométrie des espaces d’orbites avec des stabilisateurs non triviaux et le développement de nouvelles techniques pour analyser les intégrales de Bessel partielles dans ce cadre. Ces résultats ont des implications importantes pour le programme de Langlands, en particulier pour les théorèmes de réciprocité locaux et la conjecture de fonctorialité.
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Keywords: Langlands–Shahidi method, L-functions, $\epsilon $-factors, $\gamma $-factors, partial Bessel integrals
Taiwang Deng 1 ; Dongming She 1
CC-BY-ND 4.0
@article{JTNB_2025__37_2_479_0,
author = {Taiwang Deng and Dongming She},
title = {Stability of {Rankin{\textendash}Selberg} local $\gamma $-factors for split classical groups: {The} symplectic case},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {479--533},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {2},
doi = {10.5802/jtnb.1330},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1330/}
}
TY - JOUR AU - Taiwang Deng AU - Dongming She TI - Stability of Rankin–Selberg local $\gamma $-factors for split classical groups: The symplectic case JO - Journal de théorie des nombres de Bordeaux PY - 2025 SP - 479 EP - 533 VL - 37 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1330/ DO - 10.5802/jtnb.1330 LA - en ID - JTNB_2025__37_2_479_0 ER -
%0 Journal Article %A Taiwang Deng %A Dongming She %T Stability of Rankin–Selberg local $\gamma $-factors for split classical groups: The symplectic case %J Journal de théorie des nombres de Bordeaux %D 2025 %P 479-533 %V 37 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1330/ %R 10.5802/jtnb.1330 %G en %F JTNB_2025__37_2_479_0
Taiwang Deng; Dongming She. Stability of Rankin–Selberg local $\gamma $-factors for split classical groups: The symplectic case. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 479-533. doi: 10.5802/jtnb.1330
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