Strong Weil Degree Divisibility at Higher Levels

Technical Summary: Strong Weil Degree Divisibility at Higher Levels

Problem Statement
The paper investigates the arithmetic properties of morphisms from modular curves $X_0(N)$ (and intermediate curves like $X_1(N)$) to elliptic curves $E'$ in the rational isogeny class of a strong Weil curve $E$ of conductor $M$. Specifically, it addresses two primary questions regarding the relationship between the degree of the strong Weil parametrization $\pi_E: X_0(M) \to E$ and the degrees of morphisms $g: X_0(N) \to E'$ where $N$ is a multiple of $M$:

1. Divisibility: Does $\deg \pi_E$ divide $\deg g$? If not, what explicit factors control the failure of divisibility?
2. Integral Structure: Do the "old" homomorphisms induced by degeneracy maps form an integral basis for the group of homomorphisms $\text{Hom}_{\mathbb{Q}}(J_0(N), E')$? If not, can the denominators of the coefficients of integral homomorphisms be explicitly bounded?

While the case $N=M$ is well-understood via the universal property of optimal quotients, the behavior at higher levels $N > M$ involves complex interactions between degeneracy maps, Manin constants, and the structure of modular Jacobians.

Methodology
The authors employ a combination of algebraic geometry, the theory of modular forms, and lattice theory. The core methodology involves:

• Old-Form Decomposition: Utilizing the Atkin–Lehner–Li decomposition to identify the rational vector space $\text{Hom}_{\mathbb{Q}}(J_0(N), E') \otimes \mathbb{Q}$ with the span of "old" homomorphisms induced by degeneracy maps $\iota_r: X_0(N) \to X_0(M)$.
• Degree Pairings: Defining a positive-definite degree pairing on homomorphisms. This allows the degree of a morphism to be expressed as a quadratic form involving the coordinates of the homomorphism in the old basis and a "degree matrix" $A_{E,N}$.
• Denominator Bounds via Cusp Expansions: To control the denominators of the coefficients in the rational basis, the authors analyze the pullback of invariant differentials (Néron differentials) under morphisms. They establish integrality conditions by examining $q$-expansions at two specific cusps: the standard cusp $\infty$ and its image under the Fricke involution $w_N$.
• Descent Arguments: For primes $\ell$ where $\ell^2 \mid (N/M)$, the authors use a descent argument. They show that degeneracy maps $\delta_\ell: X_0(N) \to X_0(N/\ell)$ are totally ramified at $\infty$ (for $X_0$) or surjective on fundamental groups (for intermediate curves), implying that the induced map on Jacobians has a geometrically connected kernel. This property ensures that integrality descends from level $N$ to $N/\ell$.
• Tower Criterion: The authors abstract these properties into a "tower criterion" (Conditions T1–T3) to extend results from $X_0$ to compatible towers of intermediate modular curves, including $X_1$ and diamond towers.

Key Contributions and Results

1. Divisibility Theorem (Theorem 1.1):
   The paper proves that for any nonconstant morphism $g: X_0(N) \to E'$, the degree of the strong Weil parametrization divides the degree of $g$ up to a factor controlled by the Manin constant $c_E$ and the number of prime factors of $N/M$. Specifically:
   $$\deg \pi_E \mid c_E^{\Omega(N/M)} \deg g$$
   where $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity.

   • Refinement: The exponent of any prime $\ell \mid c_E$ can be sharpened from $\text{ord}_\ell(c_E)\Omega(N/M)$ to $\text{ord}_\ell(c_E) + \lfloor \text{ord}_\ell(N/M)/2 \rfloor$.
   • Unconditional Case: If $c_E = 1$ (which is known to hold when $M$ is squarefree, or verified for $M < 500,000$), then $\deg \pi_E \mid \deg g$ unconditionally.

2. Integral Basis and Denominator Theorem (Theorem 1.2):
   The authors establish conditions under which the old homomorphisms form a $\mathbb{Z}$-basis. If the Manin constant of the composite map $u \circ \pi_E$ is 1 (where $u$ generates $\text{Hom}_{\mathbb{Q}}(E, E')$), then the set $\{ \Phi_{r,u} : r \mid (N/M) \}$ forms a $\mathbb{Z}$-basis of $\text{Hom}_{\mathbb{Q}}(J_0(N), E')$.

   • General Bound: In the general case, the denominators of the coefficients of any integral homomorphism are bounded by an explicit integer $D_{u \circ \pi_E}(N/M)$, which depends on the prime factorization of $N/M$ and the Manin constant.
   • Degree Spectrum: When the Manin constant is 1, the degree of any morphism is exactly determined by the quadratic form $b^T A_{E,N} b$, where $b$ is the integer coefficient vector of the homomorphism in the old basis.

3. Extension to Intermediate Curves (Theorems 5.12–5.19):
   The results are extended to compatible towers of intermediate modular curves, including the $X_1$-tower.

   • For $X_1(N)$, analogous divisibility and basis results hold, governed by the Manin–Stevens constant $c_1$.
   • The paper demonstrates that the total ramification argument used for $X_0$ must be replaced by a topological argument involving the surjectivity of maps on fundamental groups for intermediate curves, yet the final divisibility and basis conclusions remain structurally similar.

Significance and Claims
The paper claims to provide a complete arithmetic description of the degrees of morphisms from higher-level modular curves to elliptic curves in terms of the minimal level parametrization.

• Unconditional Divisibility: It confirms that the divisibility $\deg \pi_E \mid \deg g$ is unconditional whenever $c_E = 1$, a condition satisfied for all squarefree conductors and verified computationally for a large range of conductors.
• Structural Insight: It clarifies the structure of the integral Hom group, showing that "old" maps generate the full lattice up to a controlled denominator. This resolves the question of whether the "old" lattice is saturated.
• Explicit Bounds: The paper provides explicit, computable bounds for the failure of integrality and divisibility, refining previous estimates by reducing the exponent of the Manin constant factor based on the multiplicity of prime factors in the level ratio.
• Generalizability: By formulating a "tower criterion," the authors demonstrate that these phenomena are not unique to $X_0$ but are intrinsic to compatible families of modular curves, including $X_1$.

The work relies on the semistable case of the Manin constant conjecture (proven by Česnavičius) for the unconditional results in the squarefree case and leverages existing computational verifications (Cremona) for specific conductors. The paper does not propose new applications but rather solidifies the theoretical foundation for understanding modular parametrizations at higher levels.