Elliptic curves and the residue-counts of $x^2+bx+c/x$ modulo $p$

Abstract

For any prime $p>3$ and rational $p$-integers $b,c$ with $c(b^3-27c)\not\equiv 0(\mathrm{mod}p)$ let $V_p(x^2+bx+\frac{c}{x})$ be the residue-counts of $x^2+bx+\frac{c}{x}$ modulo $p$ as $x$ runs over $1,2,\dots ,p-1$. In this paper, we reveal the connection between $V_p(x^2+bx+\frac{c}{x})$ and the number of points on certain elliptic curve over the field ${\mathbb{F}}_p$.

15 pages