Boundedness of average rank of elliptic curves ordered by the coefficients

Fatemehzahra Janbazi

REVIEW 4 major objections 3 minor 15 references

Boundedness of average rank of elliptic curves ordered by the coefficients

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Pith's one-line read Ordering elliptic curves by the coefficient height $\max(|A|,|B|)$, this paper proves the average 2-Selmer size is at most 3, hence the average rank is at most 1.5.

desk verdict Fresh method for counting quartic orbits under a non-homogeneous height, but the paper's main constant is off by 27 in the displayed volume computation and one algebraic identity does not check out. read the letter →

arxiv 2506.07089 v1 pith:Y4SHN6WE submitted 2025-06-08 math.NT

Fatemehzahra Janbazi This is my paper

classification math.NT MSC 11G0511D4511G50

keywords averagerankofellipticcurves2-Selmergroupbinaryquarticformscoefficientheightsemi-invariantsPoissonsummationequidistributionarithmeticstatistics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Every elliptic curve over the rationals is, up to a harmless normalisation, a cubic $y^2 = x^3 + Ax + B$ with integers $A$ and $B$, and this paper asks what happens to average ranks when curves are ordered by the raw coefficient height $h(E_{A,B}) = \max(|A|,|B|)$ instead of the standard height that weighs the two coefficients unevenly. The main claim is that under this thinner ordering, where the discriminant can be as large as $X^3$ and the curves of small discriminant form a zero-density subfamily, the average size of the 2-Selmer group is still at most 3, and therefore the average rank is at most 1.5. To get there the paper develops a new way to count $\mathrm{GL}_2(\mathbb{Z})$-orbits of integral binary quartic forms with bounded invariants $I$ and $J$: it replaces the five coefficients by the semi-invariants $(a, H, R, I)$, uses the syzygy $H^3 - 48a^2HI + 64a^3J = -27R^2$ to turn the constraint $|J| < X$ into an interval condition, and counts lattice points fiber by fiber via Poisson summation. The result matters because it transfers the classical average-rank bound of 1.5 to an ordering that previously available counting methods could not handle, and the technique supplies a way to count integral points inside regions that expand non-uniformly.

What carries the argument

The machine is the change of coordinates $\upsilon$ from a quartic form $(a,b,c,d,e)$ to the semi-form $(a,b,c,R,I)$, with cubic semi-invariant $R = b^3 + 8a^2d - 4abc$ and $H = 8ac - 3b^2$; for $a \neq 0$ this map is a bijection onto an explicitly described lattice $\Lambda \subset \mathbb{Z}^5$ (Lemma 2.6). Through the syzygy $H^3 - 48a^2HI + 64a^3J = -27R^2$, fixing $(a,b,c,R)$ constrains $I$ to an interval of length $O(|aX/H|)$, so the height condition $|J| < X$ becomes a one-dimensional interval condition carrying a congruence condition modulo $12a$ of the same size. Over each fiber the admissible $(R,I)$ form a lattice of covolume $|8a^2 \cdot 12a|$ whose dual Fourier transform vanishes unless $\alpha \equiv 3b\beta \pmod{12a}$ (Proposition 3.6); Poisson summation on this lattice yields the equidistribution theorem (Theorem 3.1) with per-fiber error $O(X^{1/4+12\delta_s+50\delta})$. A second fibering, over the unique rational root of a quartic form, runs the sieve: Lemma 5.4 expresses $J$ in terms of the root and the semi-invariants, and Lemma 5.5 bounds the number of such forms of height at most $X$ by $O_\epsilon(X^{7/4+\epsilon})$, giving the uniformity estimate behind Theorem 5.2.

What would settle it

Run the count in Theorem 2.2 computationally at a moderately large $X$: enumerate integral $(a,b,c)$ with $|a|,|c| \le X^{1/2}$, $|b| \le X^{1/2}$, $H = 8ac - 3b^2 \neq 0$, enumerate over each fiber the lattice points $\Lambda_{(a,b,c)}$ inside $\mathcal{H}'_{(a,b,c)}$, and compare the totals for each real-root type against $(C\zeta(2)/27)X^2$; if the difference is not $o(X^2)$, for instance if it grows like $X^{7/4}$, the central asymptotic fails. A more surgical test targets Theorem 3.1 directly: for fibers with $|H|$ near $X^{1-\delta_H}$, compute the summed Fourier error and check it truly behaves as $O(X^{1/4+12\delta_s+50\delta})$; an error of size $X^{1/2}$ per fiber, multiplied across $X^{3/2}$ fibers, would already exceed the budget.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1.1 with Corollary 1.2: when elliptic curves over $\mathbb{Q}$ are ordered by $h(E_{A,B}) = \max(|A|,|B|)$, the average size of the 2-Selmer group is at most 3 and the average rank is at most 1.5. Since rank is at most half the 2-Selmer size, the rank bound is an immediate corollary; the Selmer bound is the real target. Its engine is Theorem 2.2, an asymptotic count of irreducible integral binary quartic forms of height $h_C(f) = \max(|I(f)|, |J(f)|/C)$: there are $(C\zeta(2)/27)X^2$ orbits with 4 real roots, $(2C\zeta(2)/27)X^2$ with 2 real roots, and $(C\zeta(2)/27)X^2$ for each of the two zero-real-root families, up to $o(X^2)$. Theorem 1.4 reads the same result as constancy on average: the mean number of orbits per eligible invariant pair is $2\zeta(2)/n_i$ with $n_0 = 4$ and $n_1 = n_2 = 2$. The Selmer bound follows by sieving locally soluble orbits, using the uniformity estimate of Theorem 5.2: curves of height at most $X$ whose discriminant is divisible by $p^2$ for some $p > Q$ number $O_\epsilon(X^{2+\epsilon}/Q + X^{7/4+\epsilon})$, which keeps the sieve error below the main term. The paper states plainly that only the upper bound is proved; equality with the heuristically predicted value 3 would require a tail estimate it does not supply (Remark 4.10).

Load-bearing premise

The proof rests on exponent estimates that are, on the paper's own account, tight: the per-fiber equidistribution error $O(X^{1/4+12\delta_s+50\delta})$ of Theorem 3.1 must remain $o(X^2)$ after summing over the roughly $X^{3/2}$ fibers, and the replacement of the true height region by its simplified copy must lose at most $O_\epsilon(X^{7/4+\delta_a/2+\epsilon})$ points; if either exponent fails by a small amount, the error swallows the main term.

Editorial extensions

If this is right

The bound of 3 on the average 2-Selmer size, and of 1.5 on the average rank, holds for the coefficient-height ordering whose discriminant range reaches $O(X^3)$ — the first such result for this ordering, where the standard orbit-counting argument breaks down.
The same average-rank bound holds uniformly for every large family of elliptic curves, meaning all curves or any family cut out by finitely many congruence conditions on $A$ and $B$, under the same ordering (Theorem 5.1).
The average number of quartic orbits per eligible invariant pair is $2\zeta(2)/n_i$, a constancy-on-average statement (Theorem 1.4) that matches the heuristically predicted mean 2-Selmer size of 3, at least as an upper bound.
The paper argues that the fibering method is not specific to quartics: because semi-invariants and syzygies exist for binary $n$-ic forms generally, the same strategy may handle other orbit spaces where classical lattice-point counting performs poorly.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if, as the author expects in Remark 4.10, the upper bound is actually equality, then the random-matrix heuristic for Selmer sizes survives in a family whose discriminants reach $X^3$; that would point to a cause other than discriminant range for the heuristic failures seen in other thin families.
Beyond the paper: the core trick — replacing an Archimedean flat-projection difficulty by a congruence-modulus problem of the same size, then applying Fourier summation — reads as a general recipe for counting integral points in regions cut out by invariant inequalities, with sextic forms and higher-degree genus-one models as natural next targets.
Beyond the paper: the sieve estimate's exponent $7/4$ echoes half-integer exponents that appear in conductor-ordered counting; checking whether the two share the root-fibering mechanism would be a concrete test of whether conductor-order analogues of this average-rank bound are within reach.
Beyond the paper: the stated exponents make a checkable prediction — with $\delta$ fixed small, the summed error after fibering should be $O(X^{7/4+\epsilon})$, so a numerical experiment at $X \approx 10^8$ comparing the true count against $\zeta(2)X^2/27$ should reveal a clean $7/4$ error law if the estimates are sharp.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: this paper has a real new idea, and as written it also has a load-bearing numerical contradiction. The goal is the first average-rank bound for elliptic curves ordered by max(|A|,|B|), where the discriminant is typically O(X^3) and the usual naive-height machinery collapses. The author identifies the right obstruction—Davenport's lemma and the Bhargava-Shankar main ball fail because the coefficient height is not homogeneous—and replaces (d,e) by the semi-invariants (I,R) fiberwise over (a,b,c), then uses Poisson summation on the resulting lattice Λ_{a,b,c}. That is genuinely new and worth taking seriously. Theorems 1.3 and 1.4 are coherent and match the expected constants 2ζ(2)/n_i per eligible pair.

The soft spots are real but localized. Section 4.2 states the volume integral correctly and the prefactor Vol(F_PGL2)/(27 n_i), but the displayed conclusion drops the factor 1/27: it says N_C = 4Cζ(2)X^2/n_i instead of 4Cζ(2)X^2/(27n_i). This is not cosmetic. Theorem 2.2 and the final Selmer bound of 3 depend on that 1/27; if the true orbit count were 27 times larger, the same final computation would give an upper bound of 54 for the average of #Sel2−1, and Corollary 1.2 would not follow. I double-checked the surrounding lines: the earlier inputs in the same subsection produce the factor /27, so this is likely a typo, but it must be fixed before anyone can trust the headline theorem.

Second, Lemma 2.14 contains an algebraic mismatch. The syzygy (6) gives H^3 − 48a^2 I H + 64a^3 J = −27R^2, but the lemma rewrites this as H^3 − 3(12a)^2 I H − (12a)^3 J = R^2. Those are not equivalent: the coefficients of I and J are off, and the sign on R^2 changes. Since the proof of the O(X^{1/2+ε}) bound for reducible cubic resolvents leans on that rewritten equation to invoke BSTTTZ, the argument as written is unsupported. It may be repairable with a corrected syzygy, but a referee should not be asked to fill that in silently.

Third, the sieve input Theorem 5.2 is sketched rather than fully proved. It is plausible, and it uses a sensible embedding of cubic polynomials into quartic forms with a unique root, but the tail estimate is doing heavy lifting and needs more details. The error-term bookkeeping throughout is dense, and I did not find a separate fatal gap beyond the two algebraic issues—but the paper needs careful checking of the exponents in Section 3.

Who this is for: arithmetic statisticians and anyone working on Selmer averages or orbit counting under non-homogeneous heights. It deserves a serious referee, not a desk reject. My recommendation: send it out with major-revision expectations, explicitly asking the author to fix the Section 4.2 constant and Lemma 2.14, and to expand the proof of Theorem 5.2. Once those are addressed, the method could be an important contribution.

Referee Report

4 major / 3 minor

Summary. The paper claims that when elliptic curves E_{A,B}: y^2 = x^3 + Ax + B over Q are ordered by the coefficient height h(E_{A,B}) = max(|A|, |B|), the average size of the 2-Selmer group is at most 3, and hence the average rank is at most 1.5 (Theorems 1.1 and 1.2). The route is to count irreducible GL2(Z)-orbits of integral binary quartic forms with invariants |I| < X and |J| < CX (Theorem 2.2), using a new semi-invariant space (a,b,c,R,I), a fiberwise Poisson-summation argument, and an equidistribution statement (Theorem 3.1), followed by a volume computation (Section 4.2) and a Selmer-group argument (Section 5). The paper also states an average-per-pair version (Theorem 1.4) and a uniformity estimate for large families (Theorem 5.2).

Significance. If correct, Theorem 1.1 would be a genuinely new boundedness result: the coefficient height h(E_{A,B}) = max(|A|,|B|) has discriminant O(X^3), so the standard naive-height methods, which rely on the homogeneity of the height function, do not apply. The paper introduces a promising technique of passing to semi-invariants and reducing the Archimedean counting difficulty to a non-Archimedean congruence problem over fibers, and it explicitly identifies the role of equidistribution rather than Davenport-type lattice-point estimates. These ideas are valuable and could have applications to other thin families. However, the proof as written contains load-bearing algebraic and numerical inconsistencies that prevent verification of the main theorem: a factor-27 discrepancy in the volume computation, an incorrect syzygy rewriting in the reducibility estimate, and inconsistencies in the constants of the final Selmer average.

major comments (4)

[Section 4.2] The displayed conclusion of the volume computation, NC = 4Cζ(2)X^2/n_i + O(...), drops the factor 1/27 that is present in the immediately preceding identity NC = (Vol(F_PGL2)/(27 n_i)) ∫∫ ... . With Vol(F_PGL2) = 2ζ(2) and the computed volume integral 2CX^2, the correct conclusion is 4Cζ(2)X^2/(27 n_i), which is what Theorem 2.2 states. This is not cosmetic: the final Selmer computation in Section 5.2 cancels the orbit-count constant against local densities to obtain the Tamagawa number 2. If the true orbit count were 27 times the stated value, the same computation would give an average of #Sel2 - 1 equal to 54, and Corollary 1.2 would not follow. The factor 1/27 must be restored and all downstream constants re-checked.
[Lemma 2.14] The displayed 'rewriting' of the syzygy, H^3 - 3(12a)^2 I H - (12a)^3 J = R^2, does not follow from equation (6), which states H^3 - 48a^2 I H + 64a^3 J = -27R^2. Substituting (6) gives H^3 - 432a^2IH - 1728a^3J = -27R^2 - 384a^2IH - 1792a^3J, not R^2. The subsequent appeal to Theorems 5.1 and 5.2 of [5] is therefore unjustified, and the claimed bound O_ε(X^{1/2+ε}) for the number of classes with reducible cubic resolvent is not established. This bound is used in Lemma 2.13, equations (14)-(15), to discard reducible and non-generic forms, and hence it is load-bearing for Proposition 2.7 and Theorem 2.2.
[Theorem 1.4] Theorem 1.4 is inconsistent with Theorems 1.3 and 2.2. For example, for i=1, Theorem 1.3(b) gives ∑ h_{I,J}^{(1)} ~ 2ζ(2)/27 · X^2, while the denominator in Theorem 1.4, the number of pairs (I,J) with h(I,J)<X and Δ(I,J)<0, is asymptotic to (2+o(1))X^2 (the left half-square |I|<X, |J|<X contributes area 2X^2). The ratio is therefore ζ(2)/27, not ζ(2) as stated by the formula 2ζ(2)/n_i with n_1=2. The correct averaging constant appears to be 2ζ(2)/(27 n_i). This is a second manifestation of the missing factor 1/27 and must be corrected consistently.
[Section 5.2] In the proof of Theorem 1.1, the asymptotic for M∞(F;X) is stated as 3^4 X^2 ∏ M_p(F). But h_e(I,J) < X is equivalent to |I| < 3X and |J| < 27X, whose area is (2·3X)(2·27X) = 324X^2 = 4·3^4 X^2. The displayed count of elliptic curves is consequently off by a factor 4. Since the final bound of 2 for the average of #Sel2 - 1 is obtained by comparing the Selmer count with this curve count, the missing factor 4 would change the final average to 8 if uncorrected. The entire numerical chain in this subsection should be re-derived, especially because the transition from the orbit-count formula to the expression involving M∞(V,F;X) is not reproducible without specifying the normalization of the local and archimedean measures.

minor comments (3)

[Lemma 2.4] The statement of Lemma 2.4 bounds the number of e with |I|<X and |J|<X, but the proof uses the interval 4C/3 |aX/H| and mentions J < CX via the constant C. The statement should either include C or clarify that a fixed C is understood throughout.
[Theorem 1.4] The symbol n_i in Theorem 1.4 is defined again (n_0=4, n_1=2, n_2=2) and is not the same as the PGL2(R)-stabilizer sizes n_i introduced before Proposition 2.3, where n_{2+}=n_{2-}=4. The dual use of the same notation for different quantities is confusing and should be changed.
[Section 3.2] In Corollary 3.11 and the proof of Theorem 3.1, the parameters δ, δ_s, δ_a, δ_H, δ_e are only required to be 'sufficiently small.' Since the exponents must satisfy specific inequalities for the summed error to be o(X^2), the paper should state one explicit admissible choice of these parameters (for example, δ_s = 22δ as used later in Section 4.2) at the point where the error is bounded.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the orbit-counting and Selmer-bound arguments are derived from independent external theorems and internal estimates, not from the target result.

full rationale

The paper's central claim—an upper bound of 3 for the average size of the 2-Selmer group under coefficient-height ordering—is obtained by counting GL2(Z)-orbits of integral binary quartic forms with bounded invariants. Sections 2–4 carry out this count self-containedly: the semi-invariant parametrization, the fiberwise equidistribution theorem (Theorem 3.1), the Poisson-summation estimates, and the volume computation are all derived from stated assumptions, with no fitted constant or assumed bound on Selmer groups entering the argument. The constants ζ(2)/27 and the eventual Tamagawa number 2 are outputs of the calculation, not inputs. The paper's heavy reliance on Bhargava–Shankar [3], Bhargava–Shankar–Swaminathan [4], Shankar–Shankar–Wang [13], Shankar–Taniguchi [15], and Cremona–Fisher [9] is external support: those works are independent, established results and do not presuppose the present theorem. The author is not an author of any cited work, so there is no self-citation chain. The apparent omission of the factor 1/27 in the displayed conclusion of Section 4.2 is an arithmetic inconsistency with the preceding line and with Theorem 2.2, but it does not make the derivation circular; it is a correctness risk in exposition. Remark 4.10 notes that the tail estimate needed to upgrade Theorem 4.9 from an upper bound to an exact asymptotic is missing, but this affects sharpness only, not the circularity of the argument. No equation in the paper reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on several deep external theorems in arithmetic statistics and invariant theory, listed as axioms. There are no fitted free parameters: the constants in the main terms are universal (ζ(2) factors) and the height-family parameter C is arbitrary. No new entities are postulated; the semi-invariant space V^s is a coordinate change, not a new object.

assumptions (4)

standard math Bhargava's smooth averaging identity (Theorem 5.1 of [4]) for converting orbit sums to integrals over the fundamental domain
Invoked in Proposition 2.3 to express N(ω,L;X) as an integral; no proof is repeated.
domain assumption Counting bounds for integral points on elliptic curves (Theorems 5.1 and 5.2 of [5])
Used in Lemma 2.14 to bound the number of classes with reducible cubic resolvent by O_ε(X^{1/2+ε}).
domain assumption Embedding of minimal irreducible cubic polynomials into PGL2(Z)-orbits of quartic forms with a unique rational root (Proposition 5.3 of [13])
Basis of the tail estimate Theorem 5.2; the identities I(σ(f))=-3A and J(σ(f))=-27B are imported.
domain assumption Evaluated constants from [3]: Vol(PGL2(Z)\PGL2(R)) = 2ζ(2), local densities and Selmer mass ratios in Propositions 3.6, 3.9, 3.18 of [3]
Used to convert the weighted orbit count into the average 2-Selmer bound; these constants are external results.

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Pith. "Pith review of Boundedness of average rank of elliptic curves ordered by the coefficients." pith.science (2026). https://pith.science/paper/Y4SHN6WE

@misc{pith2026250607089,
  author       = {Pith},
  title        = {Pith review of: Boundedness of average rank of elliptic curves ordered by the coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y4SHN6WE}},
  note         = {Machine review of arXiv:2506.07089}
}

abstract

We study the average rank of elliptic curves $E_{A,B} : y^2 = x^3 + Ax + B$ over $\mathbb{Q}$, ordered by the height function $h(E_{A,B}) := \text{max}(|A|, |B|)$. Understanding this average rank requires estimating the number of irreducible integral binary quartic forms under the action of $\mathrm{GL}_2(\mathbb{Z})$, where the invariants $I$ and $J$ are bounded by $X$. A key challenge in this estimation arises from working within regions of the quartic form space that expand non-uniformly, with volume and projection of the same order. To address this, we develop a new technique for counting integral points in these regions, refining existing methods and overcoming the limitations of Davenport's lemma. This leads to a bound on the average size of the 2-Selmer group, yielding an upper bound of 1.5 for the average rank of elliptic curves ordered by $h$.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [5]

Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

Manjul Bhargava, Arul Shankar, Takashi Taniguchi, Frank Thorne, Jacob Tsimerman, and Yongqiang Zhao. Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves.arXiv preprint, January 2017. Available athttp://arxiv.org/abs/1701.02458

work page Pith review arXiv 2017

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The density of discriminants of quartic rings and fields.Ann

Manjul Bhargava. The density of discriminants of quartic rings and fields.Ann. of Math., 162:1031–1063, 2005

work page 2005

[3] [2]

The density of discriminants of quintic rings and fields.Ann

Manjul Bhargava. The density of discriminants of quintic rings and fields.Ann. of Math. (2), 172(3):1559–1591, 2010

work page 2010

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Binary quartic forms having bounded invariants, and the bound- edness of the average rank of elliptic curves.Annals of Mathematics, 181(1):191–242, 2015

Manjul Bhargava and Arul Shankar. Binary quartic forms having bounded invariants, and the bound- edness of the average rank of elliptic curves.Annals of Mathematics, 181(1):191–242, 2015

work page 2015

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Swaminathan

Manjul Bhargava, Arul Shankar, and Ashvin A. Swaminathan. The second moment of the size of the 2-selmer group of elliptic curves.arXiv preprint arXiv:2110.09063, 2021

arXiv 2021

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Birch and H.P.F

B.J. Birch and H.P.F. Swinnerton-Dyer. Notes on elliptic curves i.J. Reine Angew. Math., 212:7–25, 1963

work page 1963

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The rank of elliptic curves.Duke Mathematical Journal, 44(4):715–739, December 1977

Armand Brumer and Kenneth Kramer. The rank of elliptic curves.Duke Mathematical Journal, 44(4):715–739, December 1977

work page 1977

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Cremona.Algorithms for modular elliptic curves

J.E. Cremona.Algorithms for modular elliptic curves. Cambridge University Press, 2nd ed. edition, 2008

work page 2008