Variance of point-counts for families of cubic curves over $\mathbb{F}_p$ and Jacobsthal sums

TL;DR

This work studies the variance of the number of $\\mathbb{F}_p$-points on one-parameter families of cubic curves $y^2=f_\\lambda(x)$ by translating point counts into quadratic-character sums and evaluating their second moments. The authors develop a framework centered on Jacobsthal sums, using cubic Jacobsthal sums $\\varphi_2$ and $\\psi_3$, to obtain explicit variance formulas for several families of cubics; in many cases these variances are expressed in terms of classical character values and in others through Jacobsthal-sum evaluations tied to representations of primes as sums of squares. They present four main families, providing closed-form variances where possible and a detailed table of outcomes for a perturbation family $y^2=x^3+bx+c+\\lambda(x^2-x)$, illustrating dualities and reductions to Jacobsthal sums. Overall, the paper offers precise fluctuations of point counts across families, connecting finite-field geometry with arithmetic sums and extending prior first- and second-moment results via a Jacobsthal-sum perspective. The results deepen understanding of how arithmetic properties of $p$ influence point-count statistics on families of elliptic-type cubics over $\\mathbb{F}_p$.

Abstract

We give explicit computations for the variance of the number of points along one-parameter families of cubic curves. We highlight evaluations of variances that involve Jacobsthal sums.

Theorems & Definitions (16)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1
• Lemma 3.2
• proof : Proof of Theorem \ref{['thm: C']}
• Example 4.1
• Example 4.2
• proof : Proof of Theorem \ref{['thm: B']}