Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves

Alexander Dunn; Alexandre de Faveri; Cruz Castillo

REVIEW 2 major objections 1 minor 1 cited by

Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves

T0 review · 2 major / 1 minor · reviewed 2026-07-15 · grok-4.5

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Pith's one-line read A positive proportion of Hecke L-functions from quartic residue symbols on the Gaussian integers do not vanish at the central point, so a positive proportion of the corresponding quartic twists of y^{2}=x^{3}-x have Mordell–Weil rank zero o

desk verdict Promising positive-proportion non-vanishing for quartic Hecke L-functions over Z[i] and rank-zero for the corresponding twists, but the supplied full text is unreadable garbage so nothing can be checked. read the letter →

arxiv 2604.01316 v2 pith:VNUF6BSI submitted 2026-04-01 math.NT math.AG

Cruz Castillo , Alexandre de Faveri , Alexander Dunn This is my paper · ORCID

classification math.NTmath.AG MSC 11M4111G0511R4211F66

keywords HeckeL-functionsquarticresiduesymbolnon-vanishingMordell-WeilrankellipticcurvescongruentnumbercurvetwistsGaussianintegers

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 proves that when one takes squarefree Gaussian integers q and forms the associated quartic Hecke characters, a positive proportion of the corresponding L-functions are nonzero at the central critical point. The same analytic method applies to the Hecke characters that arise from quartic twists of the congruent number curve y^{2}=x^{3}-x. Consequently, for a positive proportion of those squarefree q ordered by norm, the twisted elliptic curve y^{2}=x^{3}-qx has Mordell–Weil rank zero over the field of Gaussian rationals. This supplies the first positive-proportion non-vanishing statement in this natural quartic family and converts it into an arithmetic statement about ranks.

What carries the argument

Mollified moments of the central values of the quartic Hecke L-functions (via approximate functional equations), which force a positive proportion of those central values to be nonzero and, through the known link for these CM twists, force the Mordell–Weil rank of E^(q) over Q(i) to be zero.

What would settle it

Compute or rigorously bound the central L-values (or the ranks of E^(q) over Q(i)) for all squarefree Gaussian q of norm up to a large X; if the proportion of non-vanishing L-values (or of rank-zero curves) tends to zero rather than remaining bounded below by a positive constant, the claim fails.

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Extended reading notes

Core claim

A positive proportion of the Hecke L-functions attached to the quartic residue symbols (·/q)_{4}, for squarefree q in Z[i], do not vanish at the central point; the method likewise yields that the elliptic curve E^(q): y^{2}=x^{3}-qx has Mordell–Weil rank 0 over Q(i) for a positive proportion of such q ordered by norm.

Load-bearing premise

The argument needs the off-diagonal contributions in the mollified second moment of the central L-values to be small enough that the main term still dominates, and it needs non-vanishing of the L-value to imply rank zero for these particular twists.

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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

The punchline is simple: the abstract states a clean, unconditional positive-proportion central non-vanishing result for the family of Hecke L-functions attached to the quartic residue symbols (·/q)_4 on squarefree q in Z[i], together with the arithmetic consequence that E^(q): y^{2} = x^{3} - qx has Mordell–Weil rank zero over Q(i) for a positive proportion of such q ordered by norm. That is a natural and worthwhile extension of the classical quadratic-twist story to the quartic setting over the Gaussian integers.

What is actually new is the move from quadratic to quartic residue symbols and the corresponding twists of the congruent-number curve, all over Q(i). The abstract frames it cleanly and without overclaim. If the proofs are there, this is solid subfield progress of the usual analytic-number-theory type—mollifiers, approximate functional equations, off-diagonal estimates—applied in a setting that has not been done before.

The soft spot is not subtle and is not manufactured: the full manuscript text we were given is almost entirely corrupted (garbled encoding, blocks of black squares or null characters) and then contaminated at the end by completely unrelated Springer material on finite-time stabilization of degenerate singular parabolic equations. None of the load-bearing analytic steps can be inspected. We therefore have no way to audit the estimates, the mollifier, the passage from non-vanishing to rank zero, or the citation pattern. That is the only real issue; the claim itself is coherent and of a familiar shape.

This paper is for people who work on non-vanishing of L-functions in families or on ranks of twists of elliptic curves over number fields. A clean source would be worth a careful look and a serious referee. As the artifact stands, I would not bring it to reading group and I would not cite it yet. A journal editor should still send a clean version to peer review rather than desk-reject; the result is important enough inside the area to deserve that time. Engage only after the authors supply a readable manuscript.

Referee Report

2 major / 1 minor

Summary. The paper claims that a positive proportion of Hecke L-functions attached to the quartic residue symbols (·/q)_4, for squarefree q in Z[i] ordered by norm, do not vanish at the central point s=1/2. The same method is said to apply to the Hecke characters arising from quartic twists of the congruent-number curve E: y^2 = x^3 - x, yielding that the twisted curves E^(q): y^2 = x^3 - qx have Mordell–Weil rank 0 over Q(i) for a positive proportion of such q. The abstract presents both statements as unconditional.

Significance. A positive-proportion central non-vanishing theorem for this natural family of degree-1 Hecke L-functions over Z[i], together with an unconditional rank-zero corollary for a positive proportion of quartic twists of y^2 = x^3 - x over Q(i), would be a solid and interesting contribution in the style of classical non-vanishing/rank results (e.g., for quadratic twists). The abstract claim is of a familiar, non-circular shape and would be of clear interest to the analytic number theory and arithmetic geometry communities if the proofs hold.

major comments (2)

The supplied full-text artifact is almost entirely unreadable: the body consists of corrupted/garbled glyphs for essentially all analytic sections, and the file is contaminated at the end by unrelated Springer material on finite-time stabilization of degenerate singular parabolic equations. Consequently the load-bearing steps—approximate functional equation for the quartic Hecke L-functions, mollifier construction and length, off-diagonal estimates, large-sieve or hybrid bounds, and the precise arithmetic input converting L(1/2, χ_q) ≠ 0 into rank_E^(q)(Q(i)) = 0—cannot be inspected or verified. No assessment of correctness is possible from the given manuscript.
Until a clean, complete version of the paper is supplied, it is impossible to determine whether the non-vanishing proportion is obtained by standard mollification over Z[i] or whether it relies on unproved hybrid subconvexity/large-sieve inputs. That distinction is load-bearing for the unconditional claim stated in the abstract, but it is not checkable here.

minor comments (1)

Only the abstract (and title/primary category) is reliably readable; even section headings and equation numbers in the body are lost to encoding corruption, so no local presentation comments can be made.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity visible: standard positive-proportion non-vanishing claim; body unreadable so no self-definitional or fitted reduction can be exhibited.

full rationale

The only readable content is the abstract, which states an unconditional positive-proportion non-vanishing result for Hecke L-functions attached to quartic residue symbols (·/q)_4 and a corresponding Mordell–Weil rank-0 statement for the quartic twists E^(q): y² = x³ − qx over Q(i). That claim is of the ordinary analytic-number-theory shape (mollified first-moment or second-moment non-vanishing for a family of degree-1 Hecke L-functions) and is not forced by defining the objects to be non-vanishing, by fitting a free parameter to the same data, or by renaming a known empirical pattern. The supplied full-text artifact is almost entirely corrupted (garbled encoding followed by unrelated Springer material on degenerate singular parabolic equations), so the approximate functional equation, mollifier, off-diagonal estimates, and the precise arithmetic input that converts L(1/2, χ_q) ≠ 0 into rank 0 cannot be inspected. Under the hard rule that circularity may be claimed only when a specific reduction can be quoted, no circular step can be recorded. The derivation, as far as it is visible, is self-contained against external benchmarks; the unreadable body is a correctness/auditability issue, not a circularity finding.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

Readable content is essentially the abstract. No free parameters or invented physical entities appear. The work rests on standard arithmetic of Z[i], Hecke L-functions for quartic residue symbols, and the link between central non-vanishing and Mordell–Weil rank for the indicated twists of y² = x³ − x. Specific unproved lemmas in the body cannot be listed because the body is unreadable.

assumptions (3)

standard math Standard analytic theory of Hecke L-functions over Q(i) for characters attached to quartic residue symbols (·/q)_4, including functional equations and approximate functional equations.
Invoked by the abstract’s non-vanishing claim for this family; standard background in the area.
domain assumption For the family E^(q): y² = x³ − qx, central non-vanishing of the associated Hecke L-function implies Mordell–Weil rank 0 over Q(i) (via the arithmetic of this CM/congruent-number setting).
The abstract treats non-vanishing and rank 0 as linked conclusions of the same method; the precise arithmetic input is standard for this curve but not checkable in the corrupted body.
domain assumption Squarefree q ∈ Z[i] ordered by norm form a well-defined asymptotic family in which positive proportion is meaningful.
Ordering and density are stated in the abstract as the setting of the main theorems.

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Cite this review

Pith. "Pith review of Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves." pith.science (2026). https://pith.science/paper/VNUF6BSI

@misc{pith2026260401316,
  author       = {Pith},
  title        = {Pith review of: Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VNUF6BSI}},
  note         = {Machine review of arXiv:2604.01316}
}

abstract

We show that a positive proportion of Hecke $L$-functions attached to the quartic residue symbols $\big( \frac{\cdot}{q} \big)_4$ for squarefree $q \in \mathbb{Z}[i]$ do not vanish at the central point. Our method also extends to the Hecke characters associated to quartic twists of the congruent number curve $E : y^2 = x^3 - x$. In particular, we prove that the elliptic curve $E^{(q)} : y^2 = x^3 - qx$ has Mordell-Weil rank $0$ over $\mathbb{Q}(i)$ for a positive proportion of squarefree $q \in \mathbb{Z}[i]$ ordered by norm.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Average analytic rank for the $L$-functions of the elliptic curves $y^2=x^3-dx$
math.NT 2026-08 conditional novelty 7.0 of 10

Under GRH the average analytic rank of y²=x³-dx over odd fourth-power-free d is at most 13/6, and at most 3/2 assuming a quartic Gauss-sum conjecture.

Reference graph

Works this paper leans on

2 extracted references · cited by 1 Pith paper

[1]

Miskolc Math

Lalvay, S., Padilla-Segarra, A., Zouhair, W.: On the existence and uniqueness of solutions for non-autonomous semi-linear systems with non- instantaneous impulses, delay, and non-local conditions. Miskolc Math. Notes23, 295–310 (2022)

2022
[2]

Milan J Math.83, 237–278 (2015)

Gal, C.G.: The role of surface diffusion in dynamic boundary conditions: Where do we stand?. Milan J Math.83, 237–278 (2015)

2015

Pith tools

Reviewed July 15, 2026 · model on record in the stance chip above.

[1] [1]

Miskolc Math

Lalvay, S., Padilla-Segarra, A., Zouhair, W.: On the existence and uniqueness of solutions for non-autonomous semi-linear systems with non- instantaneous impulses, delay, and non-local conditions. Miskolc Math. Notes23, 295–310 (2022)

2022

[2] [2]

Milan J Math.83, 237–278 (2015)

Gal, C.G.: The role of surface diffusion in dynamic boundary conditions: Where do we stand?. Milan J Math.83, 237–278 (2015)

2015