Cubic congruences and binary quadratic forms
Volume 222 / 2026
Acta Arithmetica 222 (2026), 157-190
MSC: Primary 11A07; Secondary 11A15, 11B37, 11B39, 11B50, 11E16
DOI: 10.4064/aa250326-1-10
Published online: 25 January 2026
Abstract
Let $p \gt 3$ be a prime, $a_1,a_2,a_3\in \mathbb Z$, and let $N_p(x^3+a_1x^2+a_2x+a_3)$ denote the number of solutions to the congruence $x^3+a_1x^2+a_2x+a_3\equiv 0\pmod p$. We give an explicit criterion for $N_p(x^3+a_1x^2+a_2x+a_3)=3$ via binary quadratic forms.
