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 $c(b^3-27c)\not\equiv 0\pmod p$ let $V_p(x^2+bx+\frac cx)$ be the residue-counts of $x^2+bx+\frac cx$ modulo $p$ as $x$ runs over $1,2,\ldots,p-1$. In this paper, we reveal the connection between $V_p(x^2+bx+\frac cx)$ and the number of points on certain elliptic curve over the field $\Bbb F_p$.