Elliptic curves over Fp and determinants of Legendre matrices

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, dwith d not equal 0 and c(2)-4d not equal 0, we show that there are infinitely many odd primes psuch that det [(i(2) + cij + dj(2)/p)](0 <= i,j <= p-1) = 0, where (./p) is the Legendre symbol. This confirms a conjecture of Sun. (C) 2021 Elsevier Inc. All rights reserved.