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

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 paper’s main contribution is an explicit Gross-Zagier formula relating the derivative of L-functions to the Néron-Tate heights of non-torsion CM points in Sylvester's conjecture (p ≡ 4,7 mod 9).
• It employs modular parametrizations and new CM point constructions on elliptic curves via cubic twisting to generate explicit rational points outside the classical Heegner setting.
• The derivation uses advanced harmonic analysis and explicit period computations, providing systematic insights into both Sylvester’s conjecture and the BSD conjecture.

Gross-Zagier Formula for the $4, 7$ Cases of Sylvester's Conjecture

Introduction and Background

This work addresses a central Diophantine problem dictated by Sylvester's conjecture, specifically the assertion that if $p \equiv 4,7,8 \pmod{9}$, then $p$ is a sum of two rational cubes. The negative cases $p \equiv 2,5 \pmod{9}$ are classical and do not admit nontrivial solutions. Prior breakthroughs—Elkies’ claim for the $4,7$ cases and the analytic approaches of Dasgupta-Voight for $p \equiv 4,7 \pmod{9}$ under certain cube conditions—left a gap in explicit formulae outside Heegner hypothesis configurations. This paper culminates an approach begun in [Yin26], where new CM points are constructed to affirmatively resolve Sylvester’s conjecture for $p \equiv 4,7 \pmod{9}$ for $i=1,2$.

Construction of Explicit CM Points

The elliptic curves in focus are given by $E_{p^i}: y^2 = x^3 + \frac{p^{2i}}{4}$, isogenous over $\mathbb{Q}$ to the model $p \equiv 4,7,8 \pmod{9}$0. The CM field is $p \equiv 4,7,8 \pmod{9}$1, and a splitting $p \equiv 4,7,8 \pmod{9}$2 in $p \equiv 4,7,8 \pmod{9}$3 with $p \equiv 4,7,8 \pmod{9}$4 is fixed.

Through modular parametrizations derived from Shimura’s theory, explicit CM points $p \equiv 4,7,8 \pmod{9}$5 are constructed on $p \equiv 4,7,8 \pmod{9}$6, with $p \equiv 4,7,8 \pmod{9}$7 determined by $p \equiv 4,7,8 \pmod{9}$8 when $p \equiv 4,7,8 \pmod{9}$9 and $p$0 when $p$1. These points are transported to $p$2 via isogenies arising from cubic twisting (scaling by $p$3, $p$4 components) and explicit maps. A key innovation is that the constructed points are non-torsion and defined over class extensions $p$5.

Explicit Gross-Zagier Formula

The core result is an explicit height formula of Gross-Zagier type relating the Néron-Tate height of these non-torsion CM points to the derivatives of $p$6-functions of $p$7 at $p$8. Importantly, the Gross-Zagier formula established here applies in a setting with non-self-dual automorphic representations, abelian varieties of $p$9-type, and modular parametrization that varies with $p \equiv 2,5 \pmod{9}$0. This setting stands in contrast with the Heegner point paradigm and previous work such as [HSY19], where the Heegner hypothesis was in play.

The precise formula is: $p \equiv 2,5 \pmod{9}$1 where $p \equiv 2,5 \pmod{9}$2 if $p \equiv 2,5 \pmod{9}$3, $p \equiv 2,5 \pmod{9}$4 if $p \equiv 2,5 \pmod{9}$5, $p \equiv 2,5 \pmod{9}$6 is the Néron-Tate height over the ground field, and $p \equiv 2,5 \pmod{9}$7 is the real period. It is proven that $p \equiv 2,5 \pmod{9}$8 and, hence, this factor appears in correlating the heights and derivatives for the twists.

The proofs require:

• Extensive computation of local period integrals using Whittaker models, Kirillov models, and the minimal vector method, with delicate handling of ramifications at the primes $p \equiv 2,5 \pmod{9}$9 and $4,7$0.
• Analysis of modular forms built from Hecke characters with precisely chosen conductors, accounting for subtle features of their Galois representations.
• The arithmetic of the period lattices and explicit identification of invariant differentials to obtain period relations necessary for the explicit normalization of the formula.

Theoretical and Practical Implications

Strong Claims and New Aspects:

• The formula treats cases not covered by the classical Heegner hypothesis and thus by the traditional Gross-Zagier formula.
• The explicitness in the presence of non-self-duality and for varying modular parametrizations distinguishes this formula from previous results (e.g., [Cai-Shu-Tian 2014]).
• The explicit independence of the choice of $4,7$1 in the construction and the identification of the resulting Néron-Tate height through isogenies.

Implications:

• It affirms Sylvester's conjecture for primes $4,7$2 by constructing explicit rational points of infinite order on the relevant elliptic curves.
• The explicit Gross-Zagier formula allows for the systematic computation of height pairings, supporting further numerical study of $4,7$3-values and the Birch–Swinnerton-Dyer conjecture for these curves.
• The method indicates potential generalizations for other non-Heegner settings and twists, and for other Diophantine problems involving rational points on elliptic curves with additional endomorphisms.

Connections and Future Directions

This work links the arithmetic of elliptic curves with complex multiplication, modular forms via theta series of Hecke characters, and the analytic theory of $4,7$4-functions and explicit period formulae. The explicit computation of local periods, the analysis of automorphic representations for varying isogeny classes, and the blending of adelic, modular, and Galois-theoretic methods exemplify current leading methodology in the arithmetic theory of elliptic curves.

Potential advances include:

• Extension to the remaining $4,7$5 case of Sylvester's conjecture.
• Application of these methods to related rational points on higher genus curves and their Jacobians.
• Deeper investigation of BSD predictions in the light of explicit construction and evaluation of rational points and their heights via analytic means.

Conclusion

This paper establishes an explicit Gross-Zagier formula for the $4,7$6 cases of Sylvester’s conjecture by leveraging new constructions of CM points on cubic twists of the Fermat curve. The derivation relies on advanced harmonic analysis on automorphic forms and explicit arithmetic geometry on modular and elliptic curves. The results fill a crucial gap in the literature by addressing non-classical settings where the Heegner hypothesis fails, and provide a template for further progress in explicit aspects of Diophantine geometry and the BSD conjecture.

Reference: "Gross-Zagier formula for the $4,7$7 cases of Sylvester's conjecture" (2607.01744).