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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the coefficients of Jacobi sums
in prime cyclotomic fields


Author: F. Thaine
Journal: Trans. Amer. Math. Soc. 351 (1999), 4769-4790
MSC (1991): Primary 11R18; Secondary 11T22
Published electronically: July 1, 1999
MathSciNet review: 1475696
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Abstract: Let $p\geq 5$ and $q=pf+1$ be prime numbers, and let $s$ be a primitive root mod $q$. For $1\leq n\leq p-2$, denote by $J_{n}$ the Jacobi sum $-\sum _{k=2}^{q-1}\zeta _p ^{\, \text{ind}_{s}(k)+n\, \text{ind}_{s}(1-k)}$. We study the integers $d_{n,k}$ such that $J_{n}=\sum _{k=0}^{p-1}d_{n,k}\zeta _p ^{k}$ and $\sum _{k=0}^{p-1}d_{n,k}=1$. We give a list of properties that characterize these coefficients. Then we show some of their applications to the study of the arithmetic of $\mathbb {Z} [\zeta _p +\zeta _p ^{-1}]$, in particular to the study of Vandiver's conjecture. For $m\in \mathbb {Z}-q\mathbb {Z}$, let $\rho _{n}(m)$ be the number of distinct roots of $X^{n+1}-X^{n}+m$ in $\mathbb {Z}/q\mathbb {Z}$. We show that $d_{n,k}=f-\sum _{a=0}^{f-1}\rho _{n}(s^{k+pa})$. We give closed formulas for the numbers $d_{1,k}$ and $d_{2,k}$ in terms of quadratic and cubic power residue symbols mod $q$.


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Additional Information

F. Thaine
Affiliation: Department of Mathematics and Statistics - CICMA, Concordia University, 1455, de Maisonneuve Blvd. W., Montreal, Quebec, H3G 1M8, Canada
Email: ftha@vax2.concordia.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02223-0
PII: S 0002-9947(99)02223-0
Received by editor(s): May 8, 1997
Received by editor(s) in revised form: August 29, 1997
Published electronically: July 1, 1999
Additional Notes: This work was supported in part by grants from NSERC and FCAR
Article copyright: © Copyright 1999 American Mathematical Society