Upper bounds for moments of analytic ranks of elliptic curves over number fields

Saahil Sharma; Tristan Phillips

REVIEW 3 major objections 3 minor 23 references

Upper bounds for moments of analytic ranks of elliptic curves over number fields

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read For every number field, the moments of analytic ranks of elliptic curves are bounded by (9dm/2)^m, conditionally.

desk verdict New moment bounds over number fields, but the tail theorem's proof has a divergent error term at the chosen parameters, so the advertised density exponent doesn't follow as written. read the letter →

arxiv 2607.15998 v1 pith:OKKOSENQ submitted 2026-07-17 math.NT

Tristan Phillips , Saahil Sharma This is my paper · ORCID

classification math.NT MSC 11G0511G4011M41

keywords analyticrankellipticcurvesnumberfieldsmoments1-leveldensityexplicitformulasymmetricsquareL-functionFrobeniustraces

topics Birch and Swinnerton-Dyer

open problems The Riemann Hypothesis

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

This paper establishes, under the Riemann Hypothesis for elliptic curve L-functions and their symmetric squares (plus modularity of all curves over the field), explicit upper bounds for every moment of the analytic rank in the family of elliptic curves over a number field of degree d. The m-th moment is bounded by a finite sum that is asymptotically (9dm/2)^m. As a consequence, the paper bounds the proportion of curves of analytic rank at least β, giving decay like β^{-2β/(9d)+o(β)}. These are the first such universal moment bounds over general number fields, and they unify and extend earlier conditional results for the rational field.

What carries the argument

The 1-level density D1(E/K, φ)=Σ_γ φ(γ log X/(2π)) of the low-lying zeros, together with the explicit formula expressing it as a main term plus a smoothed sum over prime ideals. A nonnegative test function φ lets the paper majorize analytic rank by D1/φ(0). The k=2 prime-power contribution is handled through the symmetric square L-function, whose possible pole at s=2 arises exactly for curves with CM defined over K. The Frobenius trace formula (Proposition 5.1) computes averages of products of traces over the height-ordered family, with odd products vanishing and even products exhibiting a leading term; the final test function φ(y)=(sin(πν y)/(2πy))^2 gives explicit constants.

What would settle it

Compute, for a number field K of degree d, the unconditionally averaged m-th power of analytic ranks up to height B and show it eventually exceeds the bound (9dm/2)^m; even a single m and K violating the bound would refute the theorem's conclusion. A practical starting point would be the first moment over an imaginary quadratic field, which the bound caps at (9d+1)/2.

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Extended reading notes

Core claim

The central discovery is a reduction of moment bounds for analytic ranks to estimates for averaged products of Frobenius traces. Using the explicit formula, analytic rank is majorized by a smoothed sum over prime ideals attached to the curve's L-function; averaging over the height-ordered family, the odd moment contributions vanish and the even contributions factor into a leading term governed by a simple integral of the chosen test function. The main term yields the normal-moment expression, and the error terms are controlled by asymptotics for the number of curves with prescribed local conditions. The paper's theorem states that, conditionally, E_K[r_an^m] ≤ sum_{k=0}^{floor(m/2)} m!/((m-2

Load-bearing premise

The entire argument requires that every elliptic curve over the arbitrary number field K be modular; for general number fields, especially non-totally-real ones, this is unproven and is a substantially stronger input than the Riemann hypothesis assumptions, so if modularity fails the explicit formula used as the starting point is not available.

Editorial extensions

If this is right

For any number field of degree d, all moments of analytic rank are finite and grow at most like (9dm/2)^m under the stated hypotheses.
The proportion of curves with analytic rank at least β decays like β^{-2β/(9d)+o(β)} for large β, faster than any fixed power of β.
The moment bound implies strong Markov-type tail decay, quantitatively improving earlier conditional estimates over the rational field.
The method works uniformly over all number fields, depending only on the degree d and the local-counting input, not on special arithmetic features of the base field.
The bounds are consistent with the minimalist philosophy that rank 0 and 1 exhaust almost all curves, though they do not prove that distribution.

Reading between the lines

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

If modularity were established for elliptic curves over arbitrary number fields, the moment bounds would become unconditional in the GRH sense; the proof's chief obstruction is the modularity input, not the analytic machinery.
The coincidence between the moment formula and the moments of a normal distribution with mean (9dm+1)/2 and variance 1/3 suggests the 1-level density may itself be asymptotically Gaussian, and a proof via Isserlis' theorem could simplify the computation.
The reliance on GRH for symmetric square L-functions could be relaxed by averaging over the family (as the paper notes), potentially yielding unconditional-in-GRH moment bounds with weaker constants.
The tail exponent -2β/(9d) implies that in high-degree number fields, large ranks remain more probable; this strong dependence on d suggests rank distribution may look quite different over large-degree fields.

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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

Two things worth knowing before you open it. The moment bound over number fields is genuinely new — it generalizes Cho–Jeong from Q and Phillips's first moment to all moments, and the PCM factorization in Lemma 3.10 is a clean way to handle CM curves. But the paper's second advertised result, the tail bound of Theorem 1.3, does not follow from the proof as written. When they take j=2m and ν=2/(9dm), the odd-part error term in Lemma 6.2 is B^{1/(3d)} log^{2m} B. Substituted into (7.5), that error term grows with B, so the equality in (7.6) is unjustified and the exponent −2β/(9d) is unsupported.

What is good: Sections 2–5 are coherent. The counting estimates are cited properly from the first author's earlier paper. The paper is honest about its conditional assumptions; the modularity assumption for arbitrary number fields is strong, but it is stated plainly and is not hidden. No fitting or constant tuning.

The soft spots are concentrated in Sections 6–7. In Theorem 6.3, the claim that odd-j terms are o(1) is false at the boundary j=m when ν=2/(9dm): Lemma 6.1 gives O(log^m B). That part is probably repairable by taking ν slightly below 2/(9dm) and then taking a limit, since the main term tends to the desired constant. The tail theorem is worse: the same limiting trick does not work, because for j=2m the error only becomes o(1) when ν < 1/(9dm), which changes the final optimization. So the advertised density bound needs a genuinely different error analysis or a revised statement.

I would not desk-reject this. The moment theorem is a real step forward and the method is worth a referee's time. But I would send it back with a request to fix the tail section. The paper is for people in arithmetic statistics and low-lying zeros; its main value is the moment framework over number fields. My recommendation: send to peer review, with clear instructions that Theorem 1.3 as stated is not justified.

Referee Report

3 major / 3 minor

Summary. The paper proves conditional upper bounds for limsup moments of analytic ranks of elliptic curves over a number field K, assuming GRH for elliptic curve L-functions and their symmetric squares, and assuming modularity of all elliptic curves over K. The proof uses an explicit formula to bound the analytic rank by a smoothed sum over prime ideals, averages the resulting prime sums over the family of elliptic curves of bounded height, and estimates the averaged sums via the first author's counting theorems for elliptic curves with prescribed local conditions. As an application, the paper claims a conditional exponential upper bound for the density of elliptic curves with analytic rank at least β.

Significance. If correct, the moment bound would generalize results of Cho–Jeong (K=Q) and the first author (first moment over number fields), with explicit, parameter-free constants. The paper is transparent about its assumptions, and the reliance on independent published counting theorems is a strength. However, the tail theorem is a separate headline claim that improves on Heath-Brown's bound, and its proof contains a load-bearing technical gap. The dependence on modularity over arbitrary number fields is also a very strong hypothesis, though it is explicitly stated.

major comments (3)

[§7, Eqs. (7.5)–(7.6)] The proof of Theorem 1.3 applies Lemma 6.2 with j=2m and ν=2/(9dm). The second error term in Lemma 6.2 is B^{3jν/2−1/(3d)} log(B)^j. Substituting j=2m and ν=2/(9dm) gives exponent 3·(2m)·(2/(9dm))/2 − 1/(3d) = 1/(3d), so the error is B^{1/(3d)} log(B)^{2m}. After the division by log(B)^{2m} in (7.5), this remains B^{1/(3d)}, which diverges as B→∞. Thus the equality in (7.6) is not justified, and Theorem 1.3 and its asymptotic (1.5) do not follow from the written derivation. This is a load-bearing error.
[§6, proof of Theorem 6.3] Immediately after (6.27), the proof asserts that for odd j, Lemma 6.1 together with ν≤2/(9dm) implies the odd-j terms are o(1). When m is odd, j=m, and ν=2/(9dm), Lemma 6.1 gives S1(m) ≪ 1 + log(B)^m, so the term in (6.27) is only O(1) after division by log(B)^m, not o(1). Consequently the asymptotic in (6.26) is not established as written for m odd. This may be repairable by taking ν strictly smaller than 2/(9dm), but the stated theorem and proof need correction.
[§3, Eqs. (3.102)–(3.108)] The passage from the pointwise bound (3.99) to the averaged binomial expansion (3.108) is not fully justified for odd m. The expansion contains terms with signs (−2)^j for odd j, and replacing S1(j) by an upper bound is only legitimate for an upper bound of the whole expression after controlling absolute values. The boundary case in Theorem 6.3 shows that this matters: an O(1) contribution from an odd j term of either sign is not negligible. The proof should either use absolute values or prove nonnegativity of the relevant combinations.

minor comments (3)

[§6, Lemma 6.2] In the proof of Lemma 6.2, the phrase 'same argument as in Proposition 6.1' refers to a non-existent proposition; it should be 'Lemma 6.1'.
[§4, after Prop. 4.2] Typo: 'de green prime ideal' should presumably be 'degree one prime ideal' or similar.
[Throughout] The notation δ_E and δ'_E is introduced with similar symbols; a short glossary would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the moment and tail bounds are derived from independent analytic and counting inputs, not from the target theorem; the apparent issue in Theorem 1.3 is a mathematical error, not circularity.

full rationale

The paper's derivation is not circular. The analytic rank is bounded using the explicit formula and the 1-level density (Prop. 3.3 and (3.17)-(3.18)), then moments of D_1 are expanded and averaged. The averaged prime-sum quantities S_1(j,phi,B) are estimated in Lemma 6.2 using the Frobenius-trace estimates of Prop. 5.1, which in turn rest on the local counting theorems of Prop. 4.1 and 4.2. At no point is the desired rank-moment bound inserted as an assumption, and no fitted constants are involved: the final constants in Theorem 1.1 come from an explicit test function phi(y)=(sin(pi nu y)/(2 pi y))^2 and the identities phi(0)=nu^2/4, integral |u|phihat(u)^2 du=phi(0)^2/6. The paper relies on the first author's earlier counting theorems [Phi25, Theorem 1.1.2, 1.1.3], but those are published, unconditional, parameter-free results whose assumptions do not include the target moment bound, so under the stated rules they constitute independent support rather than a self-citation chain. The serious weakness in the paper is not circularity: Lemma 6.2 carries an error term B^{3 j nu/2 - 1/(3d)} log(B)^j for the piece with some s_i odd, and in Theorem 1.3 at j=2m, nu=2/(9dm) this equals B^{1/(3d)} log(B)^{2m}, so the transition to (7.6) is unjustified. That is an analytic gap in the proof, not a reduction of the conclusion to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof introduces no new entities or fitted constants. It rests on standard analytic number theory, the explicit unproved hypotheses GRH and modularity, and the first author's earlier local-counting theorems. The parameter ν is a proof parameter chosen to optimize a bound, not a data-fitted value.

assumptions (5)

domain assumption GRH for L(E/K,s) and L(Sym^2 E/K,s) for all elliptic curves E over K.
Explicit formula and symmetric-square estimates in Section 3 require these unproved hypotheses; without them Proposition 3.14 fails.
domain assumption All elliptic curves over the number field K are modular.
Used for the explicit formula (Proposition 3.3) and for Gelbart–Jacquet symmetric-square functoriality. This is not known for general number fields.
domain assumption The local counting theorems of Phillips [Phi25, Thm 1.1.2, 1.1.3] and their joint extension via [Phi25, Thm 4.0.5] are valid.
Propositions 4.1, 4.2 and Remark 4.3 supply the main arithmetic input for the Frobenius trace sums; the proof does not re-derive them.
standard math Prime ideal theorem in the form Ψ(x)=log x+O_K(1) and standard Mellin/Abel summation bounds.
Used in Lemmas 3.7, 3.11, and throughout Section 6 to evaluate prime sums.
standard math Sym^2 functoriality for modular elliptic curves (Gelbart–Jacquet) and the Deuring factorization for CM elliptic curves.
Needed in Proposition 3.8 to relate S2(X) to the symmetric-square L-function and to compute δ_E.

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Cite this review

Pith. "Pith review of Upper bounds for moments of analytic ranks of elliptic curves over number fields." pith.science (2026). https://pith.science/paper/OKKOSENQ

@misc{pith2026260715998,
  author       = {Pith},
  title        = {Pith review of: Upper bounds for moments of analytic ranks of elliptic curves over number fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OKKOSENQ}},
  note         = {Machine review of arXiv:2607.15998}
}

read the original abstract

We prove a conditional upper bound for moments of the analytic rank of elliptic curves over number fields. We also prove conditional bounds for the density of elliptic curves with analytic rank larger than a given bound.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Baier, Stephan and Zhao, Liangyi , TITLE =. Adv. Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.1016/j.aim.2008.06.006 , URL =

work page doi:10.1016/j.aim.2008.06.006 2008

[2] [2]

2013 , eprint=

The average size of the 5-selmer group of elliptic curves is 6, and the average rank is less than 1 , author=. 2013 , eprint=

2013

[3] [3]

Brumer, Armand , TITLE =. Invent. Math. , FJOURNAL =. 1992 , NUMBER =. doi:10.1007/BF01232033 , URL =

work page doi:10.1007/bf01232033 1992

[4] [4]

Barquero-Sanchez, Adrian and Calvo-Monge, Jimmy , TITLE =. J. Math. Anal. Appl. , FJOURNAL =. 2025 , NUMBER =. doi:10.1016/j.jmaa.2024.129192 , URL =

arXiv 2025

[5] [5]

arXiv preprint arXiv:2506.18874 , year =

Barquero-Sanchez, Adrian and Mora-Mora, Daniel , title =. arXiv preprint arXiv:2506.18874 , year =. 2506.18874 , archivePrefix =

arXiv

[6] [6]

and Jeong, Keunyoung , TITLE =

Cho, Peter J. and Jeong, Keunyoung , TITLE =. Math. Z. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s00209-023-03362-y , URL =

work page doi:10.1007/s00209-023-03362-y 2023

[7] [7]

and Corn, Patrick and Rice, Alex and Stankewicz, James , TITLE =

Clark, Pete L. and Corn, Patrick and Rice, Alex and Stankewicz, James , TITLE =. LMS J. Comput. Math. , FJOURNAL =. 2014 , NUMBER =. doi:10.1112/S1461157014000072 , URL =

work page doi:10.1112/s1461157014000072 2014

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2005 , PAGES =

Diamond, Fred and Shurman, Jerry , TITLE =. 2005 , PAGES =

2005