高层强魏尔次数整除性
Strong Weil Degree Divisibility at Higher Levels
- Kongju National University（公州国立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究在数论领域，针对椭圆曲线的强魏尔参数化映射，证明了其次数与相关模曲线态射次数的整除关系，并将结果推广至中间模曲线的兼容塔。
AI中文摘要：
设π_E:X₀(M)→E为具有曼宁常数c_E的强魏尔参数化映射，我们研究π_E的次数与从X₀(N)（其中N是M的倍数）到E的有理同构类中椭圆曲线的态射次数之间的整除关系。我们证明，π_E的次数整除c_E的Ω(N/M)次方乘以每个此类态射的次数，其中Ω表示计重数的素因子个数。当c_E=1时，π_E的次数整除每个此类态射的次数；特别地，当M无平方因子时，根据曼宁常数猜想的半稳定情形，该整除关系无条件成立。第二个结果针对固定目标：当相关曼宁常数为1时，由退化映射诱导的旧同态构成整个Hom群的整基，且旧次数矩阵确定精确的次数谱。该论证首先表明，在将Hom群与ℚ张量积后，旧同态构成ℚ基，随后界定关于该基的整同态系数的分母。这些结果可推广至中间模曲线的兼容塔，包括X₁-塔。
英文摘要：
Let \(π_E:X_0(M)\to E\) be the strong Weil parametrization with Manin constant \(c_E\). We prove \(°π_E\mid c_E^{Ω(N/M)}°g\) for every multiple \(N\) of \(M\) and every nonconstant morphism \(g:X_0(N)\to E'\) over \(\mathbb{Q}\), where \(E'\) is \(\mathbb{Q}\)-isogenous to \(E\) and \(Ω\) counts prime factors with multiplicity. When \(c_E=1\), as is known for squarefree \(M\), the modular degree at level \(M\) therefore divides every such degree at every higher level. As an application of the divisibility theorem, we prove that no \(X_0(N)/\mathbb{Q}\) admits a morphism over \(\mathbb{Q}\) of positive odd degree at most \(1645\) to an elliptic curve of positive \(\mathbb{Q}\)-rank. For a fixed target \(E'\) and a generator \(u:E\to E'\), we also prove that if the Manin constant \(c_{u\circπ_E}=1\), the old homomorphisms induced by degeneracy maps form an integral basis of \(\operatorname{Hom}_{\mathbb{Q}}(J_0(N),E')\), and the old degree matrix determines the exact morphism degrees. The proofs bound denominators in the rational old basis. The divisibility and lattice results extend to compatible towers of intermediate modular curves, including the \(X_1\)-tower.
