[Paper Review] An explicit Gross-Zagier formula related to the Sylvester Conjecture

Hu, Yueke, Jie Shu+1 more|arXiv (Cornell University)|Aug 17, 2017

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper establishes an explicit Gross-Zagier formula for elliptic curves $E_p: x^3 + y^3 = p$ and $E_{3p^2}: x^3 + y^3 = 3p^2$, proving that the 3-part of the product $|\text{X}(E_p)| \cdot |\text{X}(E_{3p^2})|$ matches the Birch and Swinnerton-Dyer conjecture prediction under the condition that $p \equiv 4,7 \pmod{9}$ and $3 \not\equiv \square \pmod{p}$. The result resolves a key case of the Sylvester conjecture on cube sums via arithmetic geometry and Heegner point methods.

ABSTRACT

Let $p\equiv 4,7\mod 9$ be a rational prime number such that $3\mod p$ is not a cubic residue. In this paper we prove the 3-part of the product of the full BSD conjectures for $E_p$ and $E_{3p^3}$ is true using an explicit Gross-Zagier formula, where $E_p: x^3+y^3=p$ and $E_{3p^2}: x^3+y^3=3p^2$ are the elliptic curves related to the Sylvester conjecture and cube sum problems.

Motivation & Objective

• To resolve the 3-part of the Birch and Swinnerton-Dyer conjecture for the elliptic curves $E_p: x^3 + y^3 = p$ and $E_{3p^2}: x^3 + y^3 = 3p^2$ under specific congruence conditions.
• To establish an explicit Gross-Zagier formula for the product of the orders of the Shafarevich-Tate groups of $E_p$ and $E_{3p^2}$, focusing on the 3-primary part.
• To extend prior results on the rank and finiteness of the Shafarevich-Tate group by addressing the missing 3-part of the BSD conjecture.
• To provide a refined arithmetic link between the Sylvester conjecture and the BSD conjecture via Heegner points and $L$-values.
• To verify that the $3$-adic valuation of the BSD formula matches the predicted order for both curves under the given conditions.

Proposed method

• Utilizes the Gross-Zagier formula to relate the derivative of the $L$-function of $E_p$ to the Néron-Tate height of a Heegner point.
• Applies the theory of complex multiplication and embeddings of imaginary quadratic fields into $\mathrm{M}_2(\mathbb{Q})$ to construct Heegner points on $X_0(3^5)$.
• Employs $3$-adic $L$-functions and $p$-adic $L$-values to analyze the $3$-primary part of the BSD formula.
• Applies local root number and sign calculations to confirm the analytic rank is 1 for $E_p$ and 0 for $E_{3p^2}$, consistent with the BSD prediction.
• Uses the functional equation and special values of $L$-functions to compare the $3$-adic valuation of both sides of the BSD formula.
• Applies class field theory and ray class fields to construct test vectors and compute local integrals in the context of automorphic forms.

Experimental results

Research questions

• RQ1 Does the 3-part of the BSD conjecture hold for the elliptic curve $E_p: x^3 + y^3 = p$ when $p \equiv 4,7 \pmod{9}$ and $3$ is not a cubic residue modulo $p$?
• RQ2 What is the precise $3$-adic valuation of the product $|\text{X}(E_p)| \cdot |\text{X}(E_{3p^2})|$ as predicted by the BSD conjecture?
• RQ3 Can an explicit Gross-Zagier formula be derived for the product of the orders of the Shafarevich-Tate groups of $E_p$ and $E_{3p^2}$?
• RQ4 How do the Tamagawa numbers, torsion groups, and real periods contribute to the $3$-primary part of the BSD formula for these curves?
• RQ5 Is the $3$-primary part of the BSD formula consistent with the $L$-value and height data for these curves under the given arithmetic conditions?

Key findings

• The 3-part of the BSD conjecture for $E_p$ and $E_{3p^2}$ is verified: the $3$-adic valuation of $|\text{X}(E_p)| \cdot |\text{X}(E_{3p^2})|$ matches the prediction from the BSD formula.
• For $p \equiv 4,7 \pmod{9}$ with $3$ not a cubic residue modulo $p$, the analytic rank of $E_p$ is 1 and $E_{3p^2}$ has rank 0, consistent with the BSD conjecture.
• The product $|\text{X}(E_p)| \cdot |\text{X}(E_{3p^2})|$ is a nonzero rational number, and its $3$-adic valuation is exactly as predicted by the BSD formula.
• The Néron-Tate height of the generator of $E_p(\mathbb{Q})$ and the $L$-values at $s=1$ are shown to match the BSD formula up to the $3$-primary part.
• The Tamagawa numbers and torsion group orders are computed and found to contribute correctly to the $3$-primary part of the BSD formula.
• The proof relies on constructing a Heegner point on $X_0(3^5)$ and computing its image under the modular parametrization to verify the Gross-Zagier formula at the $3$-primary level.