Strong Weil Degree Divisibility at Higher Levels
Let \(\pi_E:X_0(M)\to E\) be the strong Weil parametrization with Manin constant \(c_E\). We prove \(\deg \pi_E\mid c_E^{\Omega(N/M)}\deg g\) 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 \(\Omega\) 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 \(c_{u\circ\pi_E}=1\), 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.