On new identities of Jacobi sums and related cyclotomic matrices

TL;DR

This work studies products of Jacobi sums and reveals their close ties to cyclotomic and circulant-type matrices over finite fields. By introducing the imaginary-part product $I_q( mathchi_q)$ and leveraging a variant Gauss lemma and eigenvalue techniques, the authors obtain integrality and determinant formulas that connect Jacobi sums to matrices built from quadratic characters, including $M_q(d)$ and $T_q(d)$. They prove two main theorems depending on $qmod 4$: for $q ot o 4$, an integer $x_q$ satisfies $x_q^2=2^{n-1}\det[ phi(s_i-s_j)]$, and for $q o 1mod 4$, an integer $y_q$ exists with $-a_d(q)\,y_q^2=2^n\, Det T_q(d)$, which then yields a proof of Z.-W. Sun's conjecture in this setting. The results illuminate deep connections between Jacobi sums, cyclotomic matrices, and elliptic curves over finite fields, providing exact determinant relations and concrete arithmetic consequences with potential broader impact on the study of finite-field zeta functions and related algebraic curves.

Abstract

In this paper, using some arithmetic properties of Jacobi sums, we investigate some products involving Jacobi sums and reveal the connections between these products and certain cyclotomic matrices. In particular, as an application of our main results, we confirm a conjecture posed by Z.-W. Sun in 2019.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.1
• Corollary 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof