Cubic congruences and binary quadratic forms

Abstract

Let $p>3$ be a prime, $a_1,a_2,a_3\in \mathbb{Z}$ and let $N_p(x^3+a_1{x}^2+a_{2}x+a_3)$ denote the number of solutions to the congruence $x^3+a_1{x}^2+a_{2}x+a_3\equiv 0(\mathrm{mod}p)$. In this paper, we give an explicit criterion for $N_p(x^3+a_1{x}^2+a_{2}x+a_3)=3$ via binary quadratic forms.

34 pages