Simpler congruences for Jacobi sum $J(1, 1)_{49}$ of order 49

TL;DR

The paper addresses determining congruences for Jacobi sums of order $49$ over ${\mathbb F}_p$ by building on established results for order $l^2$ and exploiting cyclotomic numbers of order $7$. The authors derive a determining congruence for $J(1,1)_{49}$ in terms of the basis $(\zeta-1)^i$ up to $i=7$, modulo $(1-\zeta)^8$, with coefficients $c_{i,1}$ expressed through solutions of a Diophantine system and Dickson-Hurwitz sums. They connect the problem to cyclotomic numbers of order $7$ and provide explicit expressions for the coefficients via $x_1,...,x_6$, along with special simplifications for artiad and hyperartiad primes. These special cases yield reduced forms of the Jacobi sum congruence, illustrating deeper arithmetic structure for primes with particular residue properties.

Abstract

In this paper we determine the congruence of Jacobi sums $J(1, 1)_{49}$ of order 49 over a field $\mathbb{F}_p$. We also show that simpler congruences hold for $J(1, 1)_{49}$ in the case of artiad and hyperartiad primes.

Theorems & Definitions (16)

• Theorem 2.1
• proof
• Lemma 4.1
• proof
• Theorem 4.1
• proof
• Lemma 6.1
• proof
• Lemma 6.2
• proof