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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.
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
We discuss the $ell$-adic case of Mazurs Program B over $mathbb{Q}$, the problem of classifying the possible images of $ell$-adic Galois representations attached to elliptic curves $E$ over $mathbb{Q}$, equivalently, classifying the rational points o
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.
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 i
We prove two theorems concerning isogenies of elliptic curves over function fields. The first one describes the variation of the height of the $j$-invariant in an isogeny class. The second one is an isogeny estimate, providing an explicit bound on th