Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun [16], Krachun and his cooperators [11] 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 $d\ne 0$ and $c^2-4d\ne 0$, we show that there are infinitely many odd primes p such that$\det {\left[\left(\frac{i^2+cij+d{j}^2}{p}\right)\right]}_{0\le i,j\le p-1}=0,$ where $(\frac{\cdot }{p})$ is the Legendre symbol. This confirms a conjecture of Sun.

中文翻译：

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Fp 上的椭圆曲线和勒让德矩阵的行列式

具有勒让德符号项的行列式与有限域上的字符和和椭圆曲线关系密切。近年来，Sun[16]、Krachun 及其合作者[11]研究了这一课题。在本文中，我们确认了 Sun 提出的一些猜想并调查了一些相关主题。

例如，给定任何整数 $C,d$ 和 $d\ne 0$ 和 $C^2-4d\ne 0$, 我们证明有无穷多个奇素数p使得$\mathrm{检测}{\left[\left(\frac{{\mathrm{一世}}^2+C\mathrm{一世}j+d{j}^2}{磷}\right)\right]}_{0\le \mathrm{一世},j\le 磷-1}=0,$ 在哪里 $(\frac{\cdot }{磷})$是勒让德符号。这证实了孙的猜想。