On the $\ell $-adic valuation of certain Jacobi sums

Vishal Arul

doi:10.5802/jtnb.1171

On the

ℓ

-adic valuation of certain Jacobi sums

Vishal Arul ¹

¹ MIT Department of Mathematics 77 Massachusetts Ave., Bldg. 2-239A Cambridge, MA 02139, USA

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 607-625.

Abstract
Résumé

Fix distinct primes $ℓ$ and $f$ , a finite field $F_{q}$ such that $q \equiv 1 (mod ℓ f)$ , multiplicative characters $χ_{ℓ}, χ_{f} : F_{q}^{\times} \to {\bar{Q}}^{\times}$ of orders $ℓ$ and $f$ , and let $J (χ_{ℓ}, χ_{f})$ be the associated Jacobi sum. We prove new a $ℓ$ -adic congruence for $J (χ_{ℓ}, χ_{f})$ . More specifically, we give a necessary and sufficient condition for $J (χ_{ℓ}, χ_{f}) \equiv - 1 (mod {(1 - ζ_{ℓ})}^{i})$ when $i \leq ℓ$ in terms of certain cyclotomic units of $F_{q}^{\times}$ being $ℓ$ th powers.

Soient $ℓ$ et $f$ deux nombres premiers distincts. Soient $F_{q}$ un corps fini tel que $q \equiv 1 (mod ℓ f)$ , $χ_{ℓ}, χ_{f} : F_{q}^{\times} \to {\bar{Q}}^{\times}$ deux caractères multiplicatifs d’ordres respectifs $ℓ$ et $f$ et $J (χ_{ℓ}, χ_{f})$ la somme de Jacobi associée. Nous prouvons une nouvelle congruence pour $J (χ_{ℓ}, χ_{f}) .$ Plus précisément, nous montrons que $J (χ_{ℓ}, χ_{f}) \equiv - 1 (mod {(1 - ζ_{ℓ})}^{i})$ avec $i \leq ℓ$ si et seulement si certaines unités cyclotomiques de $F_{q}^{\times}$ sont des puissances $ℓ$ -ièmes.

Received: 2020-07-21
Revised: 2021-01-17
Accepted: 2021-04-19
Published online: 2021-09-10

DOI: 10.5802/jtnb.1171

Classification: 11L05, 11R18, 11R27
Mots-clés : Jacobi sums, finite fields, cyclotomic units, congruences

Author's affiliations:

Vishal Arul ¹

¹ MIT Department of Mathematics 77 Massachusetts Ave., Bldg. 2-239A Cambridge, MA 02139, USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {On the $\ell $-adic valuation of certain {Jacobi} sums},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {607--625},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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     doi = {10.5802/jtnb.1171},
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Vishal Arul. On the $\ell $-adic valuation of certain Jacobi sums. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 607-625. doi : 10.5802/jtnb.1171. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1171/

References
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