On the coefficients of Jacobi sums in prime cyclotomic fields

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 $q$.