在七、八、九次数域上无限多次出现的椭圆曲线的挠群
Torsion groups of elliptic curves that appear infinitely often over septic, octic and nonic fields
AI总结：
本文确定了在七、八、九次数域上，无限多条椭圆曲线（在Q̅同构意义下）的挠群集合Φ^∞(n)，通过模曲线X_1(m,n)的低次点约化、模单位构造、有限域上属数计算及低属模曲线映射等方法证明。
摘要：
我们确定了在次数为$n=7,8,9$的数域上，无限多条椭圆曲线（在$\overline \Q$-同构意义下）作为挠群出现的阿贝尔群集合$\Phi^{\infty}(n)$。证明遵循Derickx和Sutherland对次数5和6的策略：我们将问题约化为模曲线$X_1(m,n)$上的低次点，使用模单位构造所需映射，并通过有限域上的属数计算和到低属模曲线的映射消除剩余候选。在次数$9$的曲线\(X_1(37)\)需要一个额外的论证：我们证明\(W^0_9(X_1(37))\)不包含由商映射\(X_1(37)\to X_0^+(37)\simeq 37.a1\)产生的\(J_1(37)\)的正秩椭圆因子的平移。
英文摘要：
We determine the sets $Φ^\infty(n)$ of abelian groups that appear as torsion groups of infinitely many elliptic curves, up to $\overline \Q$-isomorphism, over number fields of degree $n=7,8$ and $9$. The proof translates the problem into one about low-degree points on modular curves $X_1(m,n)$. We construct the infinite families using modular units, and eliminate the remaining candidates using finite-field gonality computations, covering arguments, and a specialization argument for $W^0_d$. The most difficult case is $X_1(37)$ in degree $9$, where the Jacobian has positive rank. We handle this case by showing that $W^0_9(X_1(37)_{\F_2})$ contains no translate of the positive-rank elliptic factor induced by the morphism $X_1(37)\to X_0^+(37)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
