Elliptic curves and the residue-counts of x²+bx+c/x modulo p

2024/07/22 by Zhi-Hong Sun, Sun, Zhi-Hong
Computer Science · Mathematics · #11E25 #11G20 #11L10 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Primary 11A15 #Secondary 11A07

paper · pdf · doi:10.48550/arxiv.2407.15432

openalex publication_date 2024/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any prime p>3 and rational p-integers b,c with c(b³-27c)\not≡ 0\pmod p let V_p(x²+bx+\frac cx) be the residue-counts of x²+bx+\frac cx modulo p as x runs over 1,2,…,p-1. In this paper, we reveal the connection between V_p(x²+bx+\frac cx) and the number of points on certain elliptic curve over the field \Bbb F_p.

Related

• Sato--Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height
• The distribution of the number of points modulo an integer on elliptic curves over finite fields
• On the equation N_p1(E)⋅ N_p2(E)⋯ N_pk(E)=n
• Elliptic curves and spin
• Power residues of Fourier coefficients of elliptic curves with complex multiplication