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arxiv: 2607.01744 · v1 · pith:DGFKITOTnew · submitted 2026-07-02 · 🧮 math.NT

Gross-Zagier formula for the 4, 7 cases of Sylvester's conjecture

Hongbo Yin This is my paper

Pith reviewed 2026-07-03 07:13 UTC · model grok-4.3

classification 🧮 math.NT
keywords Gross-Zagier formulaCM pointsSylvester's conjectureelliptic curvesL-functionsHeegner pointscanonical height
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The pith

Explicit Gross-Zagier formula proven relating CM point heights to L-function derivatives for E_{p^i}

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

The author had previously constructed CM points on the elliptic curves E_{p^i} given by y squared equals x cubed plus p to the 2i over 4 for primes p congruent to 4 or 7 modulo 9 and i equal to 1 or 2; these points produce rational solutions to x cubed plus y cubed equals p to the i and thereby settle the 4 and 7 cases of Sylvester's conjecture. This paper proves the explicit Gross-Zagier formula that equates the canonical height of these CM points to a constant multiple of the derivative at s equals 1 of the L-function attached to E_{p^i}. A sympathetic reader cares because the formula supplies a direct, computable bridge between the arithmetic size of the constructed points and the analytic behavior of the L-functions. If the relation holds, the non-torsion rational points on the cubic curves have heights that are determined by the vanishing order and derivative of the corresponding L-functions.

Core claim

We prove the explicit Gross-Zagier formula relating the height of our CM points and the derivative of the L-functions of E_{p^i}.

What carries the argument

The explicit Gross-Zagier formula that equates the Néron-Tate height of the constructed CM points on E_{p^i} to a multiple of L'(E_{p^i},1)

If this is right

  • The canonical heights of the CM points are given explicitly by the L-function derivatives.
  • The rational points on x cubed plus y cubed equals p to the i have heights determined by analytic data.
  • The formula supplies a verification of the expected height-L-function relation in these two families of curves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical checks of the formula for the smallest primes p=7 and p=13 become feasible once both sides are computed independently.
  • The same relation could be tested on any future CM points constructed for higher exponents or other congruence classes.
  • The formula links the geometric construction of rational points on the cubic surfaces directly to the analytic rank of the associated elliptic curves.

Load-bearing premise

The CM points constructed in the prior work satisfy the Heegner point conditions needed for the explicit Gross-Zagier formula to apply.

What would settle it

Direct numerical computation of the canonical height of the CM point for p=7 together with the value of L'(E_7,1); a mismatch beyond the predicted constant factor would falsify the formula.

read the original abstract

In \cite{Yin26}, the author constructed some CM points on the elliptic curves $E_{p^i}:y^2=x^3+\frac{p^{2i}}{4}$ for primes $p\equiv 4,7\mod 9$ and $i=1,2$, which give rational points on the curves $x^3+y^3=p^i$. This solves the $4,7$ cases of Sylvester's conjecture. In this paper, we prove the explicit Gross-Zagier formula relating the height of our CM points and the derivative of the $L$-functions of $E_{p^i}$.

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

1 major / 0 minor

Summary. The manuscript claims to prove an explicit Gross-Zagier formula equating the canonical height of CM points on the elliptic curves E_{p^i}: y^2 = x^3 + p^{2i}/4 (p prime ≡4 or 7 mod 9, i=1,2), constructed in the author's prior work [Yin26], to a multiple of L'(E_{p^i},1). These points are asserted to yield rational points on x^3 + y^3 = p^i, addressing the 4,7 cases of Sylvester's conjecture.

Significance. If the CM points satisfy the Heegner conditions on the appropriate modular curve, the explicit formula would furnish a concrete arithmetic-analytic relation for these specific curves, potentially confirming non-torsion points or providing evidence toward BSD in the context of the cubic Diophantine equations. The work supplies a targeted instance of Gross-Zagier but inherits its strength from the validity of the prior construction.

major comments (1)
  1. The central claim applies the explicit Gross-Zagier formula, which requires the constructed points to be Heegner points on X_0(N) (N the conductor of E_{p^i}) with matching discriminant and Atkin-Lehner eigenvalue conditions. No independent verification or explicit computation confirming these modular-curve conditions for p ≡ 4,7 mod 9 appears in the manuscript; the argument rests entirely on the construction in [Yin26] without additional checks.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript. We respond to the single major comment below.

read point-by-point responses

  1. Referee: The central claim applies the explicit Gross-Zagier formula, which requires the constructed points to be Heegner points on X_0(N) (N the conductor of E_{p^i}) with matching discriminant and Atkin-Lehner eigenvalue conditions. No independent verification or explicit computation confirming these modular-curve conditions for p ≡ 4,7 mod 9 appears in the manuscript; the argument rests entirely on the construction in [Yin26] without additional checks.

    Authors: The CM points in question are constructed in [Yin26] precisely so that they are Heegner points on the appropriate X_0(N) satisfying the required discriminant and Atkin-Lehner conditions for each E_{p^i} with p ≡ 4 or 7 mod 9. The present paper takes those points as given and derives the explicit Gross-Zagier formula relating their heights to L'(E_{p^i},1). We will revise the introduction to add an explicit sentence stating that the Heegner hypotheses are verified in [Yin26] and therefore apply here, thereby making the dependence on the prior work more transparent without repeating the full construction. revision: partial

Circularity Check

1 steps flagged

Gross-Zagier application load-bearing on self-cited CM point construction without independent Heegner verification

specific steps
  1. self citation load bearing [Abstract]
    "In \cite{Yin26}, the author constructed some CM points on the elliptic curves $E_{p^i}:y^2=x^3+\frac{p^{2i}}{4}$ for primes $p\equiv 4,7\mod 9$ and $i=1,2$, which give rational points on the curves $x^3+y^3=p^i$. This solves the $4,7$ cases of Sylvester's conjecture. In this paper, we prove the explicit Gross-Zagier formula relating the height of our CM points and the derivative of the $L$-functions of $E_{p^i}$."

    The explicit formula is asserted for these points, but their status as valid Heegner points on the modular curve X_0(N) (with matching conductor N of E_{p^i}, correct discriminant, and Atkin-Lehner eigenvalues) is justified solely by the overlapping-author citation to [Yin26]. No re-verification or explicit level/discriminant check appears in the present manuscript, so the central claim reduces to the prior self-construction.

full rationale

The paper's central result is an explicit Gross-Zagier formula equating heights of the constructed CM points to L'(E_{p^i},1). This requires the points to be Heegner points on X_0(N) satisfying the necessary discriminant, level, and Atkin-Lehner conditions for the curves E_{p^i}. The manuscript imports this status entirely via self-citation to the author's prior work [Yin26] and supplies no independent check or explicit matching of those modular conditions in the present text. While Gross-Zagier itself is an external theorem, the applicability to these specific points reduces to the self-cited construction, producing moderate circularity on the load-bearing premise.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no information on free parameters, axioms, or invented entities; all arrays are therefore empty.

pith-pipeline@v0.9.1-grok · 5629 in / 1107 out tokens · 28840 ms · 2026-07-03T07:13:15.027269+00:00 · methodology

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