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

Arul, Vishal

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

Soient $\ell$ et $f$ deux nombres premiers distincts. Soient ${\mathbf{F}}_q$ un corps fini tel que $q\equiv 1(mod\ell f)$, ${\chi }_{\ell },{\chi }_f:{\mathbf{F}}_q^{\times }\to {\overline{\mathbf{Q}}}^{\times }$ deux caractères multiplicatifs d’ordres respectifs $\ell$ et $f$ et $J({\chi }_{\ell },{\chi }_f)$ la somme de Jacobi associée. Nous prouvons une nouvelle congruence pour $J({\chi }_{\ell },{\chi }_f).$ Plus précisément, nous montrons que $J({\chi }_{\ell },{\chi }_f)\equiv -1(mod{(1-{\zeta }_{\ell })}^i)$ avec $i\le \ell$ si et seulement si certaines unités cyclotomiques de ${\mathbf{F}}_q^{\times }$ sont des puissances $\ell$-ièmes.

Fix distinct primes $\ell$ and $f$, a finite field ${\mathbf{F}}_q$ such that $q\equiv 1(mod\ell f)$, multiplicative characters ${\chi }_{\ell },{\chi }_f:{\mathbf{F}}_q^{\times }\to {\overline{\mathbf{Q}}}^{\times }$ of orders $\ell$ and $f$, and let $J({\chi }_{\ell },{\chi }_f)$ be the associated Jacobi sum. We prove new a $\ell$-adic congruence for $J({\chi }_{\ell },{\chi }_f)$. More specifically, we give a necessary and sufficient condition for $J({\chi }_{\ell },{\chi }_f)\equiv -1(mod{(1-{\zeta }_{\ell })}^i)$ when $i\le \ell$ in terms of certain cyclotomic units of ${\mathbf{F}}_q^{\times }$ being $\ell$th powers.

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

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