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arxiv: 2605.09251 · v1 · submitted 2026-05-10 · 🧮 math.NT

Recognition: no theorem link

On the coefficients of the Taylor expansion of L-functions of elliptic curves

Shuai Zhai, Tong Wei

Pith reviewed 2026-05-12 02:47 UTC · model grok-4.3

classification 🧮 math.NT
keywords elliptic curvesL-functionsTaylor expansionquadratic twistsnon-vanishingcentral valuesGRH
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The pith

Coefficients in the Taylor expansion of elliptic curve L-functions are nonvanishing for large quadratic twists under the generalized Riemann hypothesis.

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

The paper examines the coefficients appearing in the Taylor series expansion of the L-functions attached to elliptic curves over the rationals, expanded around the central point. It establishes that in the family of quadratic twists by discriminants d these coefficients do not vanish when d is large enough, assuming the generalized Riemann hypothesis for the relevant L-functions. Without that hypothesis the authors still obtain a positive lower bound on the number of nonvanishing coefficients in the family by using existing moment estimates for the central values of derivatives of the twisted L-functions. A reader cares because these coefficients determine the analytic order of vanishing at the central point and thereby influence arithmetic invariants of the curves in the twist family.

Core claim

For any elliptic curve E over Q, in the family of its quadratic twists E_d by discriminants d, the coefficients in the Taylor expansion of the L-function L(E_d, s) around s=1 are nonzero whenever d is sufficiently large, assuming the generalized Riemann hypothesis. Unconditionally, the number of such nonvanishing coefficients in the family admits a general positive lower bound derived from moment results on the central values of derivatives of quadratic twists of modular L-functions.

What carries the argument

Moment estimates for the central values of the derivatives of L-functions of quadratic twists of modular forms.

Load-bearing premise

The nonvanishing claim for large discriminants requires the generalized Riemann hypothesis to hold for the L-functions attached to the quadratic twists.

What would settle it

An explicit large discriminant d together with an elliptic curve for which one computes a vanishing Taylor coefficient in the L-function expansion would refute the conditional statement.

read the original abstract

In this paper, we investigate the coefficients of the Taylor expansion of the complex $L$-series of any elliptic curve over $\mathbb{Q}$. We prove that, in the family of quadratic twists by all the discriminants $d$, these coefficients are nonvanishing under GRH when $d$ is sufficiently large. Unconditionally, we obtain a general lower bound for the number of nonvanishing coefficients in the family of quadratic twists, through a series of results from the moments of the central values of the derivatives of quadratic twists of modular $L$-function.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper investigates the Taylor expansion coefficients of the L-functions attached to elliptic curves E over Q. It proves that, in the family of quadratic twists E_d by fundamental discriminants d, these coefficients are non-vanishing under GRH for all sufficiently large |d|. Unconditionally, it establishes a lower bound on the number of non-vanishing coefficients across the family by appealing to existing results on moments of the central values of derivatives of quadratic twists of the associated modular L-functions.

Significance. If the claims hold with the required uniformity, the work would extend non-vanishing results from the central value L(E_d,1) to higher-order Taylor coefficients in twist families. This bears on the distribution of analytic ranks and zeros in elliptic curve families and demonstrates how moment methods can be leveraged for coefficient non-vanishing beyond the leading term.

major comments (2)
  1. [Unconditional bound derivation] Abstract and the section deriving the unconditional bound: the lower bound on non-vanishing coefficients is obtained by combining moment estimates for the central values of derivatives with upper bounds on |L^{(k)}(E_d,1)|. The manuscript must explicitly cite the precise prior theorems (including any dependence on the order k) and verify that both the moment lower bounds and the upper bounds remain uniform in d for each fixed k; without this, the counting argument does not necessarily produce a positive lower bound independent of d.
  2. [GRH non-vanishing argument] The GRH section establishing non-vanishing for large |d|: the argument should clarify which specific Taylor coefficient (constant term, first derivative, etc.) is shown to be non-zero and how GRH for the twisted L-functions is used to control the location of zeros or the size of the coefficient via the functional equation or explicit formula.
minor comments (2)
  1. [Introduction and statements] Notation for the Taylor coefficients (e.g., whether they are normalized by n! or left as L^{(n)}(E,1)) should be fixed consistently throughout the introduction and statements of theorems.
  2. [References] The list of references for the invoked moment results should include the exact statements or theorems being applied, rather than a general citation to the literature on moments of quadratic twists.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which will help improve the clarity and rigor of the manuscript. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [Unconditional bound derivation] Abstract and the section deriving the unconditional bound: the lower bound on non-vanishing coefficients is obtained by combining moment estimates for the central values of derivatives with upper bounds on |L^{(k)}(E_d,1)|. The manuscript must explicitly cite the precise prior theorems (including any dependence on the order k) and verify that both the moment lower bounds and the upper bounds remain uniform in d for each fixed k; without this, the counting argument does not necessarily produce a positive lower bound independent of d.

    Authors: We agree that explicit citations and a uniformity check are required. In the revised version we will cite the precise results on moments of L^{(k)}(E_d,1) (including their k-dependence) that are invoked, and we will add a short verification paragraph confirming that, for each fixed k, the lower bounds on the moments and the upper bounds on |L^{(k)}(E_d,1)| are uniform in the discriminant d. This ensures the counting argument yields a positive lower bound independent of d, as stated. revision: yes

  2. Referee: [GRH non-vanishing argument] The GRH section establishing non-vanishing for large |d|: the argument should clarify which specific Taylor coefficient (constant term, first derivative, etc.) is shown to be non-zero and how GRH for the twisted L-functions is used to control the location of zeros or the size of the coefficient via the functional equation or explicit formula.

    Authors: We will revise the GRH section to make these points explicit. The result establishes that, under GRH, for every fixed order m the m-th Taylor coefficient of L(E_d,s) at s=1 is non-vanishing for all sufficiently large |d|. GRH places all non-trivial zeros on the critical line; combined with the functional equation (which relates the value and derivatives at s=1 to those at s=2), this prevents zeros from approaching the central point closely enough to force the m-th derivative to vanish. A brief appeal to the Hadamard product or the explicit formula then yields a uniform lower bound away from zero for large |d|. These details will be added. revision: yes

Circularity Check

0 steps flagged

No circularity: results rest on external GRH hypothesis and independent prior moment theorems

full rationale

The paper proves non-vanishing of Taylor coefficients for quadratic twists under GRH (an external named hypothesis) and derives an unconditional lower bound by invoking a series of prior results on moments of central values of derivatives of quadratic twists of modular L-functions. These moment results are treated as external inputs rather than derived or fitted inside the paper. No equation or step reduces the target coefficients or counting lower bound to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain whose validity depends on the present work. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on two external assumptions: GRH for the L-functions of the twists, and previously established moment asymptotics for the central values of their derivatives. No free parameters or new entities are introduced.

axioms (2)
  • domain assumption Generalized Riemann Hypothesis for L-functions of quadratic twists of elliptic curves
    Invoked to guarantee non-vanishing of all Taylor coefficients once d is large.
  • domain assumption Known asymptotic results for moments of central values of derivatives of quadratic twists of modular L-functions
    Used to obtain the unconditional lower bound on the number of non-vanishing coefficients.

pith-pipeline@v0.9.0 · 5383 in / 1406 out tokens · 68826 ms · 2026-05-12T02:47:55.185337+00:00 · methodology

discussion (0)

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Reference graph

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