REVIEW 3 major objections 5 minor 1 cited by
Variance of point-counts for families of cubic curves over $\mathbb{F}_p$ and Jacobsthal sums
T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash
Pith's one-line read For one-parameter families of cubic curves over finite fields, the variance of the point count is computed exactly in terms of quadratic character values and Jacobsthal sums.
desk verdict A careful, honest computation paper with genuinely new variance formulas; the main caveat is that several explicit evaluations lean on the author's forthcoming monograph without proof. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine is the translation of point counts into quadratic character sums: $\#C_\lambda = p+\sum_{x\in\mathbb{F}_p}\sigma(f_\lambda(x))$, so the variance of the point count equals the variance of the character-sum vector. This variance is computed through the identity $\sum_{\lambda}\sigma(\lambda+r(x))\sigma(\lambda+r(y))=p[x\approx y]-1$ (with Iverson brackets), which reduces the second moment to counting coincidences $r(x)=r(y)$ for a rational function $r$ associated to the family. The residual sums that cannot be evaluated by elementary character summation are cubic Jacobsthal sums $\phi_2(c)=\sum_x\sigma(x^3+cx)$ and $\psi_3(c)=\sum_x\sigma(x^3+c)$, whose values are determined by representations of $p$ as $A^2+B^2$ or $A^2+3B^2$ (Lemmas 3.1 and 3.2). These evaluations convert the variance formulas into explicit numbers.
What would settle it
For a small prime $p$, enumerate all solutions to $y^2=x^3+ax^2+bx+\lambda$ for every $\lambda\in\mathbb{F}_p$, compute the sample variance of the point counts, and compare with the formula $p-1-\sigma(-3)-\sigma(a^2-3b)$; for example $p=5$, $a=0$, $b=1$ would already settle the correctness of Theorem 2.1 for that case. Similarly, numerically evaluating the Jacobsthal sums $\phi_2(1)$ and $\psi_3(1)$ for primes up to 100 and comparing with Lemmas 3.1 and 3.2 would test those quoted identities.
Extended reading notes
Core claim
The central claim is that variance of point-counts for the families in question is not just asymptotically something but exactly computable. For instance, when $a^2\neq 3b$, the variance over $\lambda\in\mathbb{F}_p$ of $\#\{(x,y): y^2=x^3+ax^2+bx+\lambda\}$ is $p-1-\sigma(-3)-\sigma(a^2-3b)$, where $\sigma$ is the quadratic character; when $a^2=3b$ it is $(1+\sigma(-3))(p-1)$. The paper establishes analogous explicit formulas for the families $y^2=x^3+ax^2+\lambda x+c$ and $y^2=x^3+\lambda x^2+bx+c$ (Theorems 2.2 and 2.3), and for $y^2=x^3+bx+c+\lambda(x^2-x)$ it obtains a table of explicit values $\Sigma(b,c)$ for certain pairs, several in terms of the Jacobsthal sum $\phi_2(1)$ and one involving an unevaluated sum $\varrho(2)$.
Load-bearing premise
The proof depends on Jacobsthal-sum evaluations quoted without proof from a forthcoming monograph by the same author; if any of these quoted identities is incorrect, the explicit numerical values in the examples and the table in Theorem 2.4 would be wrong, although the variance formulas expressed as character sums would still hold.
Editorial extensions
If this is right
- For the family $y^2=x^3+ax^2+bx+\lambda$, the variance is completely determined by $p$, $\sigma(-3)$, and $\sigma(a^2-3b)$; no residual sums remain.
- In the twisted family $y^2=x^3+\lambda^2(bx+1)$, the variance is $p-5-\sigma(-3)-\sigma(-3b)-\phi_2(b)^2/p$, which becomes fully explicit when $p\equiv 3\pmod{4}$ or via Lemma 3.1 when $p\equiv 1\pmod{4}$.
- For $y^2=x^3+\lambda x+c$ and $y^2=x^3+\lambda x^2+c$, the residual $\psi_3$ term is evaluated by whether $c$ is a cube, giving variance in terms of $A_3,B_3$ from $p=A_3^2+3B_3^2$.
- For the dual family in Theorem 2.4, the table covers several dual pairs, recovering the Legendre family when $(b,c)=(-1,0)$ and earlier moment computations when $(-3,1)$.
- The variance formulas reproduce and generalize earlier first- and second-moment results, but from a character-sum perspective that handles families inaccessible to exponential-sum methods.
Reading between the lines
- The same variance-to-character-sum reduction should apply to one-parameter families of higher-degree curves, although computing higher moments via quadratic character sums will likely be harder; the paper explicitly notes it cannot compute moments beyond the second.
- For the family $y^2=x^3+bx+c+\lambda(x^2-x)$ with $(b,c)=(1,-1)$, the paper leaves the variance in terms of the unevaluated sum $\varrho(2)$; a testable extension is to evaluate $\varrho(2)$ using other Jacobsthal-sum identities, which would complete the table.
- Since $\sigma(-3)$ is $1$ when $p\equiv 1\pmod{3}$ and $-1$ when $p\equiv 2\pmod{3}$, Theorem 2.1 shows that for fixed $a,b$, the variance distinguishes primes by their congruence class modulo 3, an observation not highlighted in the paper.
- The explicit variance formulas give a distributional interpretation of the 'negative bias' phenomenon: for those families, the variance is slightly below $p-1$, corroborating earlier observations from a different perspective.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper computes the variance of the number of F_p-points for one-parameter families of cubic curves of the form y^2=f_lambda(x), for primes p>3. The main results are Theorem 2.1, giving the exact closed form Var = p-1-sigma(-3)-sigma(a^2-3b) (with an alternative case when a^2=3b) for the family y^2=x^3+ax^2+bx+lambda; Theorem 2.2, reducing the variance for y^2=x^3+ax^2+lambda x+c to a residual quadratic character sum and a root count n_3; Theorem 2.3, giving analogous formulas for y^2=x^3+lambda x^2+bx+c; and Theorem 2.4, providing a table of values for y^2=x^3+bx+c+lambda(x^2-x). The method rewrites #C_lambda = p+S_lambda with S_lambda a quadratic character sum, expands the second moment via sum_lambda sigma(lambda+r(x))sigma(lambda+r(y)) = p[x=y]-1, and converts the variance into counts of coincidences r(x)=r(y). Several examples evaluate residual sums with cubic Jacobsthal sums phi_2 and psi_3, yielding explicit formulas. A central limitation is that Lemma 3.1, Lemma 3.2, and several identities in Section 7 are quoted from the author's forthcoming monograph [8] and are not proved in the manuscript.
Significance. If the imported Jacobsthal-sum identities are correct, the paper gives clean, exact variance formulas that generalize earlier first- and second-moment computations of Birch, He-McLaughlin, Miller, Hopf, and Yamauchi, and it introduces a useful variance-centred framing. The core reductions in Sections 4-6 are written out in detail and are internally coherent; the counting-of-coincidences technique is elegant and yields genuinely explicit formulas for several families. The paper is honest about the residual character sums that remain unevaluated, and the examples showing how Jacobsthal sums make them explicit are valuable. The principal weakness is that the advertised explicit values in Theorem 2.4 and in Examples 5.1, 5.2, 6.1, and 6.2 depend on unproved identities from a forthcoming monograph [8]. This is a load-bearing gap, not merely a stylistic issue, because the abstract promises 'explicit computations' and the table in Theorem 2.4 advertises those values as results of the present paper.
major comments (3)
- [Section 3.2, Lemmas 3.1 and 3.2] The evaluations of the Jacobsthal sums phi_2(c) and psi_3(c) are quoted from the author's forthcoming monograph [8] with no proof. These lemmas are used directly to produce the explicit formulas in Example 5.1 (formulas (19) and (20)), Example 5.2 (formula (22)), Example 6.1 (formulas (29) and (30)), and Example 6.2 (formula (32)). Since [8] is not available to the reader, the claimed explicit computations in these examples are not verifiable from the manuscript. The paper should either prove Lemmas 3.1 and 3.2, or give a complete, accessible reference, or explicitly state that those examples are conditional on [8]. As written, the 'explicit' status of the computations is overstated.
- [Section 7, case (b,c)=(1,-1) and case (b,c)=(1,0)] The last row of the table in Theorem 2.4 depends on three unproved quotes from [8]: the palindromic quintic identity [8, Thm.5.20], the twisted symmetry [8, Thm.5.3], and the evaluation [8, Cor.5.4]. These identities are used to turn the residual palindromic quintic sum into sigma(-1)rho(2)+sigma(2)phi_2(1). Similarly, the third row (b,c)=(1,0) uses the unproved evaluation sum sigma(x^3-6x^2+x)=sigma(2)phi_2(1), cited to [8, Ch.5]. Without proofs of these identities, the values in these rows are not established by the paper. The author should include proofs or a self-contained appendix for these identities, since they are load-bearing for the advertised table.
- [Section 7, paragraph before Theorem 2.4] The paper's framing that the table in Theorem 2.4 gives 'explicit computations' is not fully accurate even after the quoted identities are accepted: the row for (b,c)=(1,-1) contains n_4 and rho(2), where rho(2) is itself an unevaluated character sum, as the paper acknowledges. More importantly, the variance formula Var #C = p-2-1/p+Sigma(b,c) is derived for general (b,c), but the table covers only seven selected pairs. The reader should be told clearly which parts of the paper are unconditional reductions and which parts are intended as explicit evaluations; the current terminology conflates the two.
minor comments (5)
- [Abstract and Introduction] The word 'explicit' is used loosely: several results, such as Theorem 2.2(ii) and Theorem 2.3(ii), are reductions to residual character sums that are not evaluated in general. Suggest clarifying the distinction between 'closed-form variance' and 'evaluation of the residual sum'.
- [Section 2, last paragraph] The claim that 'Most of our results are intractable through their exponential viewpoint' is not substantiated. It would be more appropriate to say that the author's method applies to these families, rather than asserting intractability for other methods.
- [Example 5.2, formula (22)] The signs in the four residue classes p mod 12 are not fully determined in the displayed formula: the terms '+-2A_2' and '+-2B_2' depend on the sign convention from Lemma 3.1, but the connection is not spelled out at that point. A short note would help the reader use Lemma 3.1 correctly.
- [Section 7, equation (39)] The sentence explaining why the first double sum can be extended to all of F_p^* is slightly compressed; the displayed values q(x,1)=-x and q(1,y)=-y are correct, but the reader has to reconstruct them. Expanding this one line would improve readability.
- [References] Reference [8] is listed as 'forthcoming' with no date or arXiv identifier. If the monograph is not yet available, the cited identities cannot be checked; providing a preprint link or a published volume number would be helpful.
Circularity Check
No circularity: the variance formulas are derived from elementary character-sum identities, and the cited Jacobsthal-sum evaluations are auxiliary lemmas that do not restate the target variance results.
full rationale
The central variance derivations in Theorems 2.1–2.3, and the un-evaluated part of Theorem 2.4, are self-contained. The paper rewrites #C_λ = p + S_λ, uses Var #C = Var S, and evaluates first and second moments via the elementary character identities (5)–(7), with changes of variables and diagonal counting. The Jacobsthal-sum evaluations in Lemmas 3.1–3.2 and the Section 7 identities from the author's forthcoming monograph [8] are used only to convert residual character sums into explicit numbers in the examples and in the table of Theorem 2.4. These cited identities are parameter-free statements about Jacobsthal sums and do not assume or contain the variance formulas being derived, so their use is not circular. The dependence on the unpublished monograph [8] is a verifiability and correctness risk for those explicit values, but it does not make the derivation equivalent to its inputs by construction.
Assumptions & free parameters
assumptions (4)
- standard math The standard quadratic character identities (5), (6), and (7) hold over F_p.
- domain assumption The Jacobsthal-sum evaluations stated in Lemma 3.1 and Lemma 3.2 are correct.
- domain assumption The palindromic quintic character-sum identities [8, Thm. 5.20], [8, Thm. 5.3], and [8, Cor. 5.4] are correct.
- standard math The map x to x/(x-1) is a bijection on F_p minus {0,1} for p > 3.
Cite this review
Pith. "Pith review of Variance of point-counts for families of cubic curves over $\mathbb{F}_p$ and Jacobsthal sums." pith.science (2026). https://pith.science/paper/KV5UP6MF
@misc{pith2026250415505,
author = {Pith},
title = {Pith review of: Variance of point-counts for families of cubic curves over $\mathbbF_p$ and Jacobsthal sums},
year = {2026},
howpublished = {\url{https://pith.science/paper/KV5UP6MF}},
note = {Machine review of arXiv:2504.15505}
}
read the original 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.
Forward citations
Cited by 1 Pith paper
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On quadratic character sums over quartics
The paper derives a master double-counting formula and several quartic-to-cubic transformation identities for quadratic character sums over finite fields.
Reference graph
Works this paper leans on
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Nica: Jacobsthal Sums , Monographs in Number Theory, World Scientific (forth- coming)
B. Nica: Jacobsthal Sums , Monographs in Number Theory, World Scientific (forth- coming)
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Reviewed August 16, 2026 · model on record in the stance chip above.
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