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Motivic Gauss and Jacobi sums
[Sommes motiviques de Gauss et Jacobi]
Noriyuki Otsubo1 ; Takao Yamazaki2
1Chiba University, Department of Mathematics and Informatics, Yayoicho 1-33, Inage, Chiba, 263-8522 Japan
2Chuo University, Department of Mathematics,1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551 Japan
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 1, pp. 95-135

We study the Gauss and Jacobi sums from a viewpoint of motives. We exhibit isomorphisms between Chow motives arising from the Artin–Schreier curve and the Fermat varieties over a finite field, that can be regarded as (and yield a new proof of) classically known relations among Gauss and Jacobi sums such as Davenport–Hasse’s multiplication formula. As a key step, we define motivic analogues of the Gauss and Jacobi sums as algebraic correspondences, and show that they represent the Frobenius endomorphisms of such motives. This generalizes Coleman’s result for curves. These results are applied to investigate the group of invertible Chow motives with coefficients in a cyclotomic field.

Nous étudions les sommes de Gauss et de Jacobi du point de vue des motifs. Nous démontrons des isomorphismes entre les motifs de Chow associés à la courbe d’Artin–Schreier et les variétés de Fermat sur un corps fini, qui peuvent être considérés comme (et fournissent une nouvelle preuve de) relations classiquement connues entre les sommes de Gauss et de Jacobi telles que la formule de multiplication de Davenport–Hasse. Comme étape clé, nous définissons des analogues motiviques des sommes de Gauss et de Jacobi comme des correspondances algébriques, et montrons qu’ils représentent les endomorphismes de Frobenius de tels motifs. Cela généralise le résultat de Coleman pour les courbes. Ces résultats sont appliqués à l’étude du groupe des motifs de Chow inversibles à coefficients dans un corps cyclotomique.

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Accepté le :
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DOI : 10.5802/afst.1842
Classification : 14C15, 11L05, 19E15
Keywords: Gauss sums, Jacobi sums, Davenport–Hasse relation, Chow motives, Weil numbers

Noriyuki Otsubo  1   ; Takao Yamazaki  2

1 Chiba University, Department of Mathematics and Informatics, Yayoicho 1-33, Inage, Chiba, 263-8522 Japan
2 Chuo University, Department of Mathematics,1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551 Japan
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Noriyuki Otsubo; Takao Yamazaki. Motivic Gauss and Jacobi sums. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 1, pp. 95-135. doi: 10.5802/afst.1842

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