完成有理椭圆曲线在六次域上的 torsion 子群分类
Completing the classification of torsion subgroups for rational elliptic curves over sextic fields
AI总结：
该研究证明了有理椭圆曲线在六次域上不存在 C_3 ⊕ C_{18} torsion 子群，通过 Galois 表示理论和 Diophantine 几何方法完成分类。
摘要：
我们完成了有理椭圆曲线 E/Q 在六次数域 K 上 torsion 子群 E(K)_{tors} 的分类。先前的研究已经确定了完整的这些群集合，仅剩下 C_3 ⊕ C_{18} 的存在性存疑。我们证明该群不存在。我们的证明依赖于椭圆曲线附带的 Galois 表示理论。假设存在 C_3 ⊕ C_{18} torsion 子群将对曲线的 mod-2 和 3-adic Galois 表示施加强烈的同步约束。通过应用最近对椭圆曲线 over Q 的 ℓ-adic Galois 图像的分类，我们将这些算术约束转化为一个 Diophantine 几何问题：这样的曲线的 j-不变量必须对应于有限多个模曲线上的有理点。我们然后利用经典方法分析这些曲线，证明其中没有对应于无复乘的椭圆曲线所需的有理点，从而证明我们的主要结果。
英文摘要：
We complete the classification of torsion subgroups $E(K)_{\text{tors}}$ that can occur for an elliptic curve $E/\mathbb{Q}$ over a sextic number field $K$. Previous work determined the complete set of these groups, leaving the existence of only one group in question: $C_3 \oplus C_{18}$. We prove that this group does not occur. Our proof relies on the theory of Galois representations attached to elliptic curves. The assumed existence of a $C_3 \oplus C_{18}$ torsion subgroup would impose strong, simultaneous constraints on the mod-$2$ and $3$-adic Galois representations of the curve. By applying the recent classification of $\ell$-adic Galois images for elliptic curves over $\mathbb{Q}$, we translate these arithmetic constraints into a problem of Diophantine geometry: the $j$-invariant of such a curve must correspond to a rational point on one of the finitely many modular curves. We then analyze these curves using classical methods and show that none have the necessary rational points corresponding to elliptic curves without complex multiplication, thereby proving our main result.
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