Jacobi sums and Stickelberger's congruence
Let \(p\) be a prime, \(q = p^f\) and \(\zeta_p\), \(\zeta_{q - 1}\) roots of unity of order \(p\) and \(q - 1\) respectively. For any prime \({\mathfrak p}\) in \(\mathbb{Q} (\zeta_{q - 1})\) over \(p\), let \({\mathfrak P}\) be the unique prime in \(\mathbb{Q} (\zeta_p, \zeta_{q - 1})\) over \({\mathfrak p}\). Let \(\omega_{\mathfrak p}\) denote the Teichmüller character on \(\mathbb{F}_q \cong \mathbb{Z} [\zeta_{q - 1}]/{ \mathfrak p}\). For \(0 \leq a < q - 1\), write the base \(p\) expansion of \(a\) as \(a = a_0 + a_1 p + \cdots + a_{f - 1} p^{f - 1}\) where \(0 \leq a_i \leq p - 1\) (not all \(a_i = p - 1)\). Fix \(\zeta_p\) and define the Gauss sum relative to a multiplicative character \(\chi\) of \(\mathbb{F}_q\) by \(G (\chi) = - \sum_{x \in \mathbb{F}_q} \chi (x) \zeta_p^{\text{Tr}_{\mathbb{F}_q/ \mathbb{F}_p} (x)}\). The Jacobi sum relative to the multiplicative characters \(\chi_1\), \(\chi_2, \dots, \chi_r\) of \(\mathbb{F}_q\) is defined by \(J (\chi_1, \dots, \chi_r) = (- 1)^{(r - 1)} \sum \chi_1 (x_1) \cdots \chi_r (x_r)\) where the sum is taken over all \(r\)-tuples \((x_1, \dots, x_r) \in \mathbb{F}^r_q\) such that \(x_1 + x_2 + \cdots + x_r = 1\). Theorem 1. (Stickelberger) With the above notation in force, one has \[ G (\omega_{\mathfrak p}^{- a}) \equiv {(\zeta_p - 1)^{a_0 + \cdots + a_{f - 1}} \over a_0! \cdots a_{f-1}!} \pmod {{\mathfrak P}^{a_0 + \cdots + a_{f - 1} + 1}}. \] Theorem 2. For \(r \geq 1\) and \(0 \leq k_1 \leq \cdots \leq k_r < q - 1\) with some \(k_j > 0\), \[ J (\omega_{\mathfrak p}^{- k_1}, \dots, \omega_{\mathfrak p}^{- k_r}) \equiv {(k_1 + \cdots + k_r)! \over k_1! \cdots k_r!} \pmod {\mathfrak p}. \] In this paper, Theorem 2 is proved using elementary \(p\)-adic analysis. Then Theorem 1 is shown to be equivalent to Theorem 2. The formula in Theorem 1 is valid also for characteristic 2, thereby extending the earlier work of \textit{N. Koblitz} and \textit{B. Gross} [Ann. Math., II. Ser. 109, 569-581 (1979; Zbl 0406.12010)] in the case of odd characteristic.
scientific article; zbMATH DE number 819171