用弗罗贝尼乌斯迹推导椭圆曲线的最小韦伊斯特拉斯系数，揭示其与同构类的关系。
Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves
- 通过决策树分析，从素数2、3的弗罗贝尼乌斯迹可精确恢复前两个系数。
- 加入导子奇偶性后，第三个系数也可完全恢复，且公式为新发现。
- 结果表明前三系数由同构类决定，适用于代数数论与密码学研究者。
我们研究了有理数域上椭圆曲线的约化最小韦伊斯特拉斯系数能否由其弗罗贝尼乌斯迹确定。决策树模型显示，仅需素数2和3处的弗罗贝尼乌斯迹，即可完美恢复前两个系数；若再补充导子奇偶性，则第三个系数亦可恢复。随后，我们给出了基于弗罗贝尼乌斯迹与导子奇偶性的显式公式，这些公式似乎是全新的。特别地，我们得出结论：椭圆曲线的前三个约化最小韦伊斯特拉斯系数由其同构类唯一确定。
原文摘要 · Abstract (English)
We investigate the extent to which the reduced minimal Weierstrass coefficients of an elliptic curve over $\mathbb{Q}$ may be computed from it's Frobenius traces. Decision tree models reveal that the first two reduced minimal Weierstrass coefficients can be recovered with perfect accuracy from the Frobenius traces at the primes $2$ and $3$, and the third by supplementing these two traces with the conductor parity. We subsequently prove explicit formulae for these coefficients using the Frobenius traces and conductor parity. These formulae appear to be new. In particular, we deduce that the first three reduced minimal Weierstrass coefficients of an elliptic curve are determined by its isogeny class.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。