Legendre Symbol of $\prod f(i,j) $ over $ 0<i<j<p/2, \ p\nmid f(i,j) $

TL;DR

This paper studies the Legendre symbol of products of linear and quadratic forms over finite fields, providing explicit evaluations and unified identities, extending previous work by Sun.

Contribution

It offers new explicit formulas and identities for Legendre symbol products of forms, including linear and quadratic cases, with conditions for evaluation and connections to previous conjectures.

Findings

Explicit evaluation of Legendre symbol products for quadratic forms.

Unified identities for products involving linear forms.

Explicit results for specific parameter values like k=2,4,5,9,10.

Abstract

Let $p>3$ be a prime. We investigate Legendre symbol of $\displaystyle \prod_{0<i<j<p/2 \atop p\nmid f(i,j) } f(i,j) \ $, where $i,j\in \Bbb Z, f(i,j)$ is a linear or quadratic form with integer coefficients. When $f=ai^2+bij+cj^2$ and $p\nmid c(a+b+c)$ , we prove that to evaluate the product is equivalent to determine $ \displaystyle \sum_{y=1}^{p-1} \bigg(\frac{y(y+1)(y+k)}{p}\bigg) \pmod{16}$ , where $4c(a+b+c)k \equiv (4ac-b^2)\pmod{p}.$ Parallel results are given for $\displaystyle \prod_{i,j=1 \atop p\nmid f(i,j) }^{(p-1)/2} \bigg(\frac{ f(i,j) }{p}\bigg).$ Then we show that $ \displaystyle \sum_{y=1}^{p-1} \bigg(\frac{y(y+1)(y+k)}{p}\bigg) \pmod{16}$ can be evaluated explicitly when k=2,4,5,9,10 or k is a square. And for several classes of f(i,j) these two kinds of products can be evaluated explicitly. Finally when f is a linear form we give unified identities for these products. Thus we prove these kind of problems raised in Sun's paper.