Binary quadratic forms and elliptic curves with analytic rank one

Shuai Zhai; Tong Wei

REVIEW 3 major objections 4 minor 15 references

Binary quadratic forms and elliptic curves with analytic rank one

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper proves that, under mild root-number and coprimality assumptions, infinitely many coprime pairs (u,v) make the twisted elliptic curve Q(u,v)y^2=f(x) have analytic rank one.

desk verdict The intended extension is the right problem and the analytic route is plausible, but the statement and the sum actually estimated don't match, and the fixed-form claim doesn't follow from a genus-weighted mean. read the letter →

arxiv 2607.18728 v1 pith:W65GYRVE submitted 2026-07-21 math.NT

Tong Wei , Shuai Zhai This is my paper · ORCID

classification math.NT MSC 11G0511G4011E1611F67

keywords ellipticcurvesquadratictwistsanalyticrankonebinaryformsgenustheoryL-functionderivativesweightedmeanvaluescongruentnumbers

topics Birch and Swinnerton-Dyer

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

The paper aims to show that, for an arbitrary elliptic curve E:y^2=f(x) over Q and any positive definite binary quadratic form Q(u,v) whose discriminant is coprime to the conductor (with an additional root-number condition when the conductor is a square), infinitely many coprime integer pairs (u,v) make the twisted curve Q(u,v)y^2=f(x) have analytic rank one. The route is a weighted mean value theorem: the sum over squarefree d of r_Q(d)L'(1/2,E^(d))F(d/X), where r_Q(d) counts representations of d by the forms in the genus of Q, equals a nonzero constant times X log X plus a smaller error. A nonzero main term that dominates the error forces infinitely many of the derivatives to be nonzero, hence infinitely many twists of analytic rank one. The argument combines genus-theoretic representation weights, an approximate functional equation, reciprocity for quadratic characters, and a second-moment bound on quadratic twists of the associated modular L-function. If correct, the result lifts a phenomenon previously known for elliptic curves with complex multiplication to the general case.

What carries the argument

The central object is the genus representation weight r_Q(d)=∏_{i=1}^h(ψ_i(d)+ε_i)∏_{p|d}(1+χ_D(p)), which is nonzero exactly when d is represented by some form in the genus of Q. The proof inserts the factor (1+χ_{-4}(d)) and an auxiliary character χ_D so that only d≡1 mod 4 contribute; for such d, quadratic reciprocity gives χ_d(n)=χ_n(d). That converts the d-sum into a product of Dirichlet L-functions whose only pole in the region sits at s=1; its residue produces the αX log X main term. The error terms are controlled by truncating the n-sum at Y=X/(log X)^{100}, a dyadic partition of unity, and a second-moment estimate for quadratic twists of the modular L-function, with a residue at the

What would settle it

For a concrete positive definite binary quadratic form Q of discriminant D whose genus has more than one class, compute all squarefree d up to large X with r_Q(d)>0 using formula (2.1). If the weighted average in Theorem 1.2 has a nonzero main term while no (or only finitely many) of those d are of the form Q(u,v) with gcd(u,v)=1, the bridge from the mean value to Theorem 1.1 fails; observing infinitely many such d represented by the genus but not by Q itself would refute the claimed implication.

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Extended reading notes

Core claim

The core claim is Theorem 1.1: under the assumptions (D,q)=1, and when q is a square also assuming some twist has root number -1, there are infinitely many coprime pairs (u,v) such that the elliptic curve E_{u,v}: Q(u,v)y^2=f(x) has analytic rank one, meaning L'(1/2,E_{u,v}) is nonzero. The engine is Theorem 1.2, a weighted average over squarefree d coprime to 2qD: the sum of r_Q(d)L'(1/2,E^(d))F(d/X) has main term αX log X with α≠0 whenever q is not a square or the root number of E is -1. Because log X grows faster than (log X)^{1/2}(log log X)^3, the nonzero main term leaves no room for all the derivatives to vanish, so infinitely many d in the genus-represented set must give nonvanishing

Load-bearing premise

The load-bearing premise is that a nonzero genus representation count r_Q(d) can be taken to mean d=Q(u,v) for some coprime u,v; this is automatic only when the genus of Q contains a single class, and the paper supplies no transfer for general class number—the step appears where Theorem 1.1 is said to follow from Theorem 1.2 and where Q is replaced by the principal form in the proof.

Editorial extensions

If this is right

Infinitely many d in the set represented by the genus of Q satisfy L'(1/2,E^(d))≠0, so the corresponding twisted curves have analytic rank one.
The nonzero main term rules out the possibility that all central derivatives vanish in the weighted family, so the nonvanishing twists form an infinite set.
For a positive definite binary quadratic form of odd discriminant, this yields infinitely many congruent numbers properly represented by the form, via the congruent-number elliptic curve.
The conditions on q and the root number are used only to make the leading coefficient α nonzero; when q is a square and the root number is +1, the argument is not claimed to apply.

Reading between the lines

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

A lower-bound version of the mean value, rather than an asymptotic, might give a positive proportion of rank-one twists among genus-represented d's; the present argument only forces infinitude because the genus weight r_Q(d) can be large and irregular.
The assertion that the coprimality assumption (D,q)=1 is removable is stated but not carried out in the written proof; a direct extension would trace the local factors at primes dividing q through the same residue computation.
The main-term residue is expressed through a symmetric-square L-function, so the constant α is in principle computable; a numerical check for a small conductor and a small discriminant would test the predicted X log X term.

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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

Don't cite this as a theorem yet; the paper's central claim is not proved as written. The target—extending Munshi's CM-family result to arbitrary elliptic curves via nonlinear quadratic twists—is the right problem, and coupling it to Li's second-moment bound is a natural approach. You can see a serious analytic attempt in the main-term residue computation and the error-term structure.

But there are two load-bearing gaps. First, Theorem 1.2 states a weighted average of L'(1,E^{(d)}), while the proof at (3.1) computes a sum of L'(1/2, f⊗χ_Dχ_d), an auxiliary twist family. No argument connects these L-functions or their nonvanishing sets. Second, even if you grant that sum, a positive genus-weighted mean (r_Q(d) from (2.1)) only forces d to be represented by some form in the genus of Q. For class number >1, that does not imply d=Q(u,v) for the fixed form in Theorem 1.1. The proof makes it worse by silently replacing Q with the principal form of discriminant D. Note the abstract is actually weaker than Theorem 1.1—'genus of Q' versus fixed Q—and the paper never decides which claim it is proving.

What is good: the authors take a real step in the literature, use the right recent inputs (Li, Munshi, Zhou), and the main-term computation is standard enough to be plausible. The applications, e.g., to generalized congruent numbers, are worth stating.

What is missing: a precise statement of the sum the proof actually establishes, a bridge between E^{(d)} and the auxiliary twist, and a clear statement of whether the result concerns the fixed form or the genus. Several key estimates are imported from [10] and [15] without stating the lemmas or checking whether they hold for non-CM newforms. That may be a fixable exposition issue, but the other two are logical.

If this comes to your journal, don't desk-reject; the intended result is significant enough to referee, but the referee should be sent to check the statement-to-proof match and the genus-to-form transfer. As it stands, I would send back for major revision, not accept. Reading group: maybe—useful for seeing how to check that the family in a moment computation matches the family in the theorem.

Referee Report

3 major / 4 minor

Summary. The paper claims that for an elliptic curve E: y^2 = f(x) and a positive definite binary quadratic form Q of discriminant D with (D,q)=1, there are infinitely many coprime integer pairs (u,v) such that the curve Q(u,v)y^2 = f(x) has analytic rank one. This is presented as a consequence of a weighted first-moment theorem (Theorem 1.2) over squarefree integers represented by the genus of Q. The proof follows Munshi's method, using the approximate functional equation, a Dirichlet-series main term, and recent second-moment results of Li and Zhou. The core of the paper is the derivation of the weighted moment in Section 3, split into a main-term proposition (3.1) and an error-term proposition (3.2).

Significance. If established, the result would be a substantial advance: it would extend Munshi's nonlinear quadratic-twist nonvanishing from CM elliptic curves to arbitrary elliptic curves and would have applications to generalized congruent number problems. The proposed strategy, combining Munshi's method with Li's second-moment bounds, is plausible and the paper identifies the right types of estimates. However, the central logical connections are missing: the moment actually computed in Section 3 is not the moment stated in Theorem 1.2, and Theorem 1.1 does not follow from a genus-weighted average. Because these gaps concern the main claim rather than presentation, they cannot be considered minor.

major comments (3)

[§3, Eq. (3.1)] The sum actually evaluated in Section 3 is Σ (1+χ_{-4}(d)) r(d) L'(1/2, f⊗χ_Dχ_d) F(d/X), whereas Theorem 1.2 requires Σ r_Q(d) L'(1,E^{(d)}) F(d/X). With the normalization of Section 2, L'(1,E^{(d)}) corresponds to L'(1/2, f⊗χ_d), not to L'(1/2, f⊗χ_Dχ_d). The inserted character χ_D is not an auxiliary 'throwing in' — it changes the quadratic-twist family. Moreover, the approximate functional equation used in (3.3)–(3.4) has sign factor (1 - ε_f χ_d(-q*)), not (1 - ε_f χ_Dχ_d(-q*)); thus the proof is not even a consistent computation for f⊗χ_Dχ_d. Since Theorem 1.2 is the engine from which Theorem 1.1 is derived, this mismatch is load-bearing.
[§2, (2.1); §3, first paragraph] Theorem 1.2 is a genus-weighted average: r_Q(d)>0 is equivalent to d being represented by some form in the genus of Q, not necessarily by Q itself. Theorem 1.1, however, requires d=Q(u,v) for the fixed form Q. At the start of Section 3, Q is additionally replaced by the principal form of discriminant D. No argument is supplied that positivity of the genus-weighted mean transfers to the fixed principal form, nor that the resulting d satisfy Q(u,v)=d. When a genus contains more than one inequivalent class, the principal form represents only a subset of the integers represented by the genus. Thus, even if Propositions 3.1 and 3.2 were fully correct, Theorem 1.1 does not follow.
[§3.1.2, bound for U(N,t)] The proof of Proposition 3.1 relies on the estimate U(N,t) ≪ δ(t) N (log N)^{3/2 - θ}, quoted from [10, Lemma 5]. Munshi's paper [10] is explicitly about CM elliptic curves, and no argument is given that this second-moment bound holds for the arbitrary weight-2 newform f introduced in Section 2. This estimate is used to bound the error term E and is therefore needed for the claimed asymptotic. The reference to [10] is not by itself sufficient unless the lemma is known to be valid in the non-CM case; the authors need to state and justify this extension.

minor comments (4)

[§2, Eq. (2.1)] The function r_Q(d) is described as the number of representations of d by forms in the genus of Q, but the displayed formula is a local indicator-type expression. Please clarify whether r_Q(d) is a 0/1 indicator or an actual representation count; the proof seems to use only the indicator property.
[§3.2, Eqs. (3.12)–(3.13)] Theorem 1.2 states that F is an arbitrary nonnegative smooth compactly supported function, but (3.12) defines a special dyadic F. The authors should explain how the general case follows, or restate Theorem 1.2 with the specific F actually used.
[§3.3, Eqs. (3.3) and (2.5)] Equation (2.5) uses the smoothing W(n/|d|), while A(d) in (3.3) uses W(n/Y). The role of B(d) in correcting this discrepancy should be stated explicitly at the beginning of Section 3; currently the decomposition appears abruptly.
[§3.1.2 and §3.2] The symbol H is used both for the dyadic weight H(N) in (3.10) and for an integer in (3.12). This is confusing and should be adjusted.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; the analytic mean-value result is self-contained and externally supported. The fixed-form Theorem 1.1 is a logical overreach from the genus-wide Theorem 1.2, not a circular reduction.

full rationale

The paper's central estimate, Theorem 1.2, is an honest weighted first moment: the left-hand side is a genuine genus-weighted sum r_Q(d)L'(1,E^{(d)}), and the main term alpha X log X arises from contour integration, residues, and a factorization involving L(s, sym^2 f). There is no fitted parameter that is later renamed as a prediction, and no definition of the target quantity is smuggled into the input weight. The proof uses external theorems and methods: Munshi [10] for the nonlinear-family approach, Li [8] and Zhou [15] for second-moment bounds, and standard genus theory. These are independent supports, not self-citations. The only self-citation is [14], which appears in a list of subsequent developments and is not used in any proof step; it is therefore non-load-bearing. A separate defect exists but is not circular: Theorem 1.1 is asserted to follow from Theorem 1.2, yet r_Q(d) counts representations by all forms in the genus of Q, not necessarily by the fixed form Q. For class number greater than one, a positive genus-weighted mean value only forces infinitely many d in the genus set, which is exactly the abstract's claim. This is a logical gap or overstatement in the statement of Theorem 1.1, not a self-referential derivation. Accordingly, the circularity score is 2, reflecting only the minor non-load-bearing self-citation; the derivation itself is not circular.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted parameters or invented entities. The load-bearing axioms are imported external estimates; the most fragile is the application of Munshi's CM lemma to non-CM forms. The silent reduction of the given Q to the principal form is an unflagged assumption that affects the derived theorem.

assumptions (7)

standard math E over Q is modular: there exists a primitive weight-2 newform f of level q with L(s+1/2,E)=L(s,f).
Invoked in Section 2; for elliptic curves over Q this is the modularity theorem, accepted background.
standard math Deligne bound |λ_f(n)| ≤ τ(n).
Used throughout the estimates, including the approximate functional equation and dyadic blocks.
domain assumption Genus theory formula (2.1) and the implication (1+χ_{-4}(d))r(d)≠0 ⇒ r_Q(d)≠0.
Used to pass from representation counts to the weight in (3.1)-(3.2). Assumes the given Q can be replaced by the principal form of the same discriminant.
ad hoc to paper Munshi's Lemma 5 bound U(N,t)≪δ(t)N(logN)^{3/2−θ} holds for arbitrary non-CM weight-2 newforms.
The proof exports a CM-specific estimate to the non-CM case without derivation; see Section 3.1.2.
domain assumption Zhou's [15, Lemma 7.1/Section 7.5] and Li's [8] second-moment estimates apply in the dyadic decomposition of Proposition 3.2.
These results are external and are invoked as black boxes; no verification is given.
domain assumption Assumptions (D,q)=1 and, if q is a square, the existence of a pair (u0,v0) with root number −1.
Stated in the Introduction and used throughout; without them the main term or the even/odd twist conditions differ.
standard math The dyadic partition of unity function G satisfying (3.11) exists.
Standard in analytic number theory; the paper cites Warner [13] and Li [8, p.709].

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Cite this review

Pith. "Pith review of Binary quadratic forms and elliptic curves with analytic rank one." pith.science (2026). https://pith.science/paper/W65GYRVE

@misc{pith2026260718728,
  author       = {Pith},
  title        = {Pith review of: Binary quadratic forms and elliptic curves with analytic rank one},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W65GYRVE}},
  note         = {Machine review of arXiv:2607.18728}
}

abstract

Given an elliptic curve with Weierstrass equation $y^2=f(x)$, and a positive definite binary quadratic form $Q(u, v)$. We show that there are infinitely many $d$ in the set represented by the quadratic forms in the genus of $Q$ such that the twisted elliptic curve $dy^2=f(x)$ has analytic rank one.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 3 linked inside Pith

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Pith tools

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

[1] [10]

Munshi, R.,On quadratic families of CM elliptic curves, Transactions of the American Math- ematical Society363(8) (2011), 4337–4358. 1, 2, 8

2011

[2] [15]

2, 9 Tong Wei,Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, Shandong, China

Zhou, Z.,Moment of Derivatives of Quadratic Twists of ModularL-Functions, arXiv:2503.14680v2 (2025). 2, 9 Tong Wei,Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, Shandong, China. E-mail:202421344@mail.sdu.edu.cn Shuai Zhai,Mathematical Research Center, Shandong University, Jinan, Shandong, China. E-mail:zhai@...

arXiv 2025

[3] [1]

Bump, D., Friedberg, S., Hoffstein, J.,Eisenstein series on the metaplectic group and non- vanishing theorems for automorphicL-functions and their derivatives, Ann. of Math. (2)131 (1990), 53–127. 1

1990

[4] [2]

153–193, Progr

Goldfeld, D., Hoffstein, J., Patterson, S.,On automorphic functions of half-integral weight with applications to elliptic curves, Number theory related to Fermat’s last theorem (Cambridge, Mass., 1981), pp. 153–193, Progr. Math., 26, Birkh¨ auser, Boston, Mass., 1982. 1

1981

[5] [3]

Huang T.,Averaging quadratically twistedL-values and their derivatives, arXiv:2507.00179 (2025). 2

arXiv 2025

[6] [4]

Huang, T., Wei, Z.,First moment of derivatives ofL-functions in a nonlinear family, arXiv:2607.09056v2 (2026). 2

arXiv 2026

[7] [5]

American Mathematical Society Colloquium Publications, Rhode Island (2004)

Iwaniec, H., Kowalski, E.,Analytic Number Theory, vol.53. American Mathematical Society Colloquium Publications, Rhode Island (2004). 2

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Jiang, Y., Shen, Q., Tang, Z.,The second moment of derivatives of quadratic twists of modular L-functions, arXiv:2603.21739v1 (2026). 2

arXiv 2026