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Register a new userElliptic curves over $mathbb{F}_p$ and determinants of Legendre matrices
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Authors
Hai-Liang Wu
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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 $d e0$ and $c^2-4d e0$, we show that there are infinitely many odd primes $p$ such that $$detbigg[left(frac{i^2+cij+dj^2}{p}right)bigg]_{0le i,jle p-1}=0,$$ where $(frac{cdot}{p})$ is the Legendre symbol. This confirms a conjecture of Sun.
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We study an analogue of Serres modularity conjecture for projective representations $overline{rho}: operatorname{Gal}(overline{K} / K) rightarrow operatorname{PGL}_2(k)$, where $K$ is a totally real number field. We prove new cases of this conjecture when $k = mathbb{F}_5$ by using the automorphy lifting theorems over CM fields established in previous work of the authors.
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The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some variants. For example, we show the following result: Let $p=a^2+4b^2$ be a prime with $a,b$ integers and $aequiv1pmod4$. Then for the determinant $$S(1,p):={rm det}bigg[left(frac{i^2+j^2}{p}right)bigg]_{1le i,jle frac{p-1}{2}},$$ the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.
Let $mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $mathcal{R}=mathbb{F}_p+umathbb{F}_p+u^2mathbb{F}_p+u^{3}mathbb{F}_{p}+cdots+u^{k}mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem.
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