Strong Weil Degree Divisibility at Higher Levels

Abstract

Let $\pi_E:X_0(M)\to E$ be the strong Weil parametrization with Manin constant $c_E$. We study divisibility relations between the degree of $\pi_E$ and the degrees of morphisms from $X_0(N)$, where $N$ is a multiple of $M$, to elliptic curves in the rational isogeny class of $E$. We prove that the degree of $\pi_E$ divides $c_E^{\Omega(N/M)}$ times the degree of every such morphism, where $\Omega$ counts prime factors with multiplicity. When $c_E=1$, the degree of $\pi_E$ divides the degree of every such morphism. In particular, the divisibility is unconditional when $M$ is squarefree, by the semistable case of the Manin constant conjecture. A second result concerns a fixed target. When the relevant Manin constant is one, the old homomorphisms induced by degeneracy maps form an integral basis of the full Hom group, and the old degree matrix determines the exact degree spectrum. The argument first shows that the old homomorphisms form a $\mathbb{Q}$-basis after tensoring the Hom group with $\mathbb{Q}$ and then bounds the denominators of the coefficients of integral homomorphisms with respect to this basis. These results extend to compatible towers of intermediate modular curves, including the $X_1$-tower.