Sylvester猜想的证明

中文摘要 AI 辅助

我们证明了Sylvester猜想，该猜想源于他1879年对三元三次方程的研究，即每个素数$p\equiv4,7,8\pmod9$都是两个有理数的立方和。Elkies在1994年宣布了对类$4$和$7$的证明，Yin最近给出了完整的证明。对于剩余的类$p\equiv8\pmod9$，我们证明了椭圆曲线$E_p:y^2=x^3+p^2/4$具有解析秩一，正如Birch和Swinnerton-Dyer猜想所预测的那样，因此$p$是两个有理数的立方和。证明首先改编了作者在CM椭圆曲线秩一逆定理工作中使用的辅助Rankin--Selberg构造。所得的卷积分解为$E_p$的$L$-函数乘以一个互补的$L$-函数。Chan的$3$-同源下降和秩零逆定理表明互补的中心$L$-值非零，因此只需证明相关的Heegner点的三次分量是非挠的。一个基本困难是底层CM轨道的未加权Hecke迹消失。我们的决定性想法是在取迹之前进行$\lambda$-除法，其中$\lambda=1-\omega$且$\omega$是原始三次单位根。我们通过分析$p$处的Frobenius证明了所得的除法边界非零。CM轨道上的Galois作用和分歧理论然后将这种非消失性转移到三次分量上。

英文摘要

We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime $p\equiv4,7,8\pmod9$ is a sum of two rational cubes. Elkies announced a proof for the classes $4$ and $7$ in 1994, and Yin recently supplied a complete proof. For the remaining class $p\equiv8\pmod9$, we prove that the elliptic curve $E_p:y^2=x^3+p^2/4$ has analytic rank one, as predicted by the Birch and Swinnerton-Dyer conjecture, and so $p$ is a sum of two rational cubes. The proof begins by adapting the auxiliary Rankin--Selberg construction from the authors' work on the rank one converse for CM elliptic curves. The Rankin--Selberg $L$-function factors as the $L$-function of $E_p$ times a complementary $L$-function. Chan's $3$-isogeny descent and the rank zero converse show that the complementary central $L$-value is non-zero, and so it suffices to prove that a cubic component of the associated Heegner point is non-torsion. A basic difficulty is that the unweighted Hecke trace of the underlying CM orbit vanishes. Our decisive idea is to take $λ$-division before taking the trace, where $λ=1-ω$ and $ω$ is a primitive cube root of unity. We prove that the resulting division boundary is non-zero by analysing Frobenius at $p$. The Galois action on the CM orbit and ramification theory then transfer this non-vanishing to the cubic component.

发表机构

• The University of Texas at Austin（德克萨斯大学奥斯汀分校）
• Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）

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