In the analytic study of trace functions of -adic sheaves over finite fields, a crucial issue is to control the conductor of sheaves constructed in various ways. We consider cohomological transforms on the affine line over a finite field which have trace functions given by linear operators with an additive character of a rational function in two variables as a kernel. We prove that the conductor of such transforms is bounded in terms of the complexity of the input sheaf and of the rational function defining the kernel, and discuss applications of this result, including motivating examples arising from the Polymath8 project.
Dans l’étude analytique des fonctions traces de faisceaux –adiques sur les corps finis, un problème crucial est de contrôler la taille du conducteur de faisceaux construits de façons variées. Nous considérons les transformées cohomologiques sur la droite affine sur un corps fini dont les fonctions traces sont données par des opérateurs linéaires dont la matrice est un caractère additif évalué sur une fonction rationnelle en deux variables. Nous prouvons que le conducteur de telles transformées est borné en fonction du conducteur du faisceau de départ et de la fraction rationnelle définissant le noyau. Enfin nous présentons des applications et des exemples, en particulier des exemples provenant du projet Polymath8.
Accepted:
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Keywords: Étale cohomology, conductor, $\ell $-adic sheaves, Riemann Hypothesis over finite fields, exponential sums
@article{AFST_2021_6_30_1_203_0, author = {Fouvry, \'Etienne and Kowalski, Emmanuel and Michel, Philippe}, title = {On the conductor of cohomological transforms}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {203--254}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 30}, number = {1}, year = {2021}, doi = {10.5802/afst.1671}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/afst.1671/} }
TY - JOUR AU - Fouvry, Étienne AU - Kowalski, Emmanuel AU - Michel, Philippe TI - On the conductor of cohomological transforms JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2021 SP - 203 EP - 254 VL - 30 IS - 1 PB - Université Paul Sabatier, Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1671/ DO - 10.5802/afst.1671 LA - en ID - AFST_2021_6_30_1_203_0 ER -
%0 Journal Article %A Fouvry, Étienne %A Kowalski, Emmanuel %A Michel, Philippe %T On the conductor of cohomological transforms %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2021 %P 203-254 %V 30 %N 1 %I Université Paul Sabatier, Toulouse %U http://archive.numdam.org/articles/10.5802/afst.1671/ %R 10.5802/afst.1671 %G en %F AFST_2021_6_30_1_203_0
Fouvry, Étienne; Kowalski, Emmanuel; Michel, Philippe. On the conductor of cohomological transforms. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 1, pp. 203-254. doi : 10.5802/afst.1671. http://archive.numdam.org/articles/10.5802/afst.1671/
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