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/

[1] Vishal Arul Torsion points on Fermat quotients of the form y n =x d +1 (2020) (https://arxiv.org/abs/1910.14251)

[2] Bruce C. Berndt; Ronald J. Evans; Kenneth S. Williams Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, 1998 | Zbl

[3] Keith Conrad Jacobi sums and Stickelberger’s congruence, Enseign. Math., Volume 41 (1995) no. 1-2, p. 141-141 | MR | Zbl

[4] Ronald J. Evans Congruences for Jacobi sums, J. Number Theory, Volume 71 (1998) no. 1, pp. 109-120 | DOI | MR | Zbl

[5] Yasutaka Ihara Profinite braid groups, Galois representations and complex multiplications, Ann. Math., Volume 123 (1986) no. 1, pp. 43-106 | DOI | MR | Zbl

[6] Kenkichi Iwasawa A note on Jacobi sums, Symposia Math, Volume 15 (1975), pp. 447-459 | Zbl

[7] Tomasz Jędrzejak On the torsion of the jacobians of superelliptic curves y q =x p +a, J. Number Theory, Volume 145 (2014), pp. 402-425 | DOI | MR | Zbl

[8] Tomasz Jędrzejak A note on the torsion of the jacobians of superelliptic curves y q =x p +a, Algebra, logic and number theory (Banach Center Publications), Volume 108, Polish Academy of Sciences, 2016, pp. 143-149 | MR | Zbl

[9] Nicholas M. Katz Crystalline Cohomology, Dieudonné Modules, and Jacobi Sums, Automorphic forms, representation theory and arithmetic (Tata Institute of Fundamental Research Studies in Mathematics), Volume 10, Springer, 1981, pp. 165-246 | DOI | Zbl

[10] Hiroo Miki On the -adic expansion of certain Gauss sums and its applications, Galois representations and arithmetic algebraic geometry (Advanced Studies in Pure Mathematics), Volume 12, North-Holland, 1987, pp. 87-118 | DOI | MR | Zbl

[11] Tsuyoshi Uehara On a congruence relation between Jacobi sums and cyclotomic units, J. Reine Angew. Math., Volume 382 (1987), pp. 199-214 | MR | Zbl

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