On the -adic valuation of certain Jacobi sums
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 607-625.

Fix distinct primes and f, a finite field F q such that q1(modf), multiplicative characters χ ,χ f :F q × Q ¯ × of orders and f, and let J(χ ,χ f ) be the associated Jacobi sum. We prove new a -adic congruence for J(χ ,χ f ). More specifically, we give a necessary and sufficient condition for J(χ ,χ f )-1(mod(1-ζ ) i ) when i in terms of certain cyclotomic units of F q × being th powers.

Soient et f deux nombres premiers distincts. Soient F q un corps fini tel que q1(modf), χ ,χ f :F q × Q ¯ × deux caractères multiplicatifs d’ordres respectifs et f et J(χ ,χ f ) la somme de Jacobi associée. Nous prouvons une nouvelle congruence pour J(χ ,χ f ). Plus précisément, nous montrons que J(χ ,χ f )-1(mod(1-ζ ) i ) avec i si et seulement si certaines unités cyclotomiques de F q × sont des puissances -ièmes.

Published online:
DOI: 10.5802/jtnb.1171
Classification: 11L05,  11R18,  11R27
Keywords: Jacobi sums, finite fields, cyclotomic units, congruences
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},
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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/

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