Strong Weil Degree Divisibility at Higher Levels

Abstract

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.