论文标题
$ \ mathbb {f} _p $和legendre矩阵的决定因素上的椭圆曲线
Elliptic curves over $\mathbb{F}_p$ and determinants of Legendre matrices
论文作者
论文摘要
具有Legendre符号条目的决定因素与有限字段的字符总和和椭圆曲线密切相关。近年来，Sun，Krachun和他的合作者研究了这个话题。在本文中，我们确认了Sun提出的一些猜想，并研究了一些相关主题。 例如，给定任何整数$ c，d $带有$ d \ ne0 $和$ c^2-4d \ ne0 $，我们表明有很多奇数的奇数$ p $，因此$ \ det \ det \ bigg [\ weft（\ frac {i^2+cij+cij+cij+cij+dj^2}}} {p} {p i i \ p-1} = 0，$$其中$（\ frac {\ cdot} {p}）$是legendre符号。这证实了太阳的猜想。
Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures posed by Sun and investigate some related topics. For instance, given any integers $c,d$ with $d\ne0$ and $c^2-4d\ne0$, we show that there are infinitely many odd primes $p$ such that $$\det\bigg[\left(\frac{i^2+cij+dj^2}{p}\right)\bigg]_{0\le i,j\le p-1}=0,$$ where $(\frac{\cdot}{p})$ is the Legendre symbol. This confirms a conjecture of Sun.
