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

TL;DR

This paper explores the relationship between the residue counts of a specific rational quadratic polynomial modulo a prime and the number of points on related elliptic curves over finite fields.

Contribution

It establishes a novel connection between polynomial residue counts and elliptic curve point counts over finite fields.

Findings

Residue counts are linked to elliptic curve point enumeration.

Provides formulas relating polynomial residues to elliptic curve data.

Enhances understanding of elliptic curves via residue distribution analysis.

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 $\Bbb F_p$.