On the Finiteness of Isolated $j$-invariants for $X_1(N)$

Technical Summary: On the Finiteness of Isolated j-Invariants for X1(N)

Problem Statement
The classification of rational points on modular curves $X_1(N)$ is a central problem in arithmetic geometry. While points belonging to infinite parameterized families (such as those arising from maps to $\mathbb{P}^1$ or abelian varieties) are well-understood, "isolated" points—those not belonging to such families—present a significant obstruction to a complete classification of points of a fixed degree. This paper focuses specifically on the collection of isolated $j$-invariants for $X_1(N)$, defined as the $j$-values of isolated points on $X_1(N)$ mapped to the $j$-line $X_1(1) \cong \mathbb{P}^1$.

The primary question addressed is Question 1 (from Bourdon et al. [10]): Are there only finitely many isolated $j$-invariants of each fixed degree? While Merel's Uniform Boundedness Theorem guarantees finitely many isolated points of a fixed degree on any specific $X_1(N)$, it does not immediately imply finiteness of isolated $j$-invariants as $N$ varies. This paper investigates the relationship between this question and other uniformity conjectures in the field and establishes new finiteness results, particularly for elliptic curves with rational $j$-invariants.

Methodology
The paper employs a combination of moduli theory, Galois representation analysis, and geometric bounds on modular curves.

1. Hypothesis Interplay: The author establishes logical implications between four major hypotheses concerning the uniformity of elliptic curves over number fields of fixed degree $d$:

   • Hypothesis 1: Generalized Serre Uniformity (surjectivity of $p$-adic Galois representations for large $p$).
   • Hypothesis 2: Finiteness of isolated $j$-invariants of degree $d$.
   • Hypothesis 3: Non-CM Isogeny Bounds (finiteness of non-cuspidal, non-CM points of degree $d$ on $X_0(N)$ for large $N$).
   • Hypothesis 4: Refined Polynomial Bounds on torsion growth.
     The paper proves that Hypothesis 1 implies Hypothesis 2, and crucially, that Hypothesis 2 implies both Hypothesis 3 and Hypothesis 4. This positions Question 1 as a refinement of the isogeny and torsion bounds that lacks the full strength of generalized Serre uniformity.

2. Galois Representation and Entanglement: To address the rational case ($d=1$), the paper analyzes the image of the mod $p$ and $p$-adic Galois representations $\rho_{E,p}$ and $\rho_{E,p^\infty}$. It utilizes classification results for these images (e.g., Mazur, Serre, Bilu, Parent, Rebolledo, Lemos) to determine the degree of points on $X_1(N)$. A key technical tool is the analysis of "entanglement" between torsion fields of distinct primes. The author derives new lower bounds on the degree of points on $X_1(N)$ by controlling the ramification and the contribution of primes $p$ where the image is not surjective (specifically non-split Cartan subgroups).

3. Gonality and Isolation: The paper utilizes lower bounds on the gonality of modular curves (Abramovich) and the relationship between the degree of a point and the genus of the curve. If the degree of a point exceeds the genus (or specific gonality bounds), the point cannot be isolated.

Key Contributions and Results

• Implications Among Hypotheses (Theorem 9): The paper formally proves that the finiteness of isolated $j$-invariants (Hypothesis 2) is a sufficient condition to establish both the finiteness of non-CM points on $X_0(N)$ for large levels (Hypothesis 3) and polynomial bounds on torsion growth (Hypothesis 4).
• Finiteness for Rational $j$-invariants (Theorem 3): The main unconditional result establishes that there are only finitely many rational isolated $j$-invariants associated with modular curves of the form $X_1(p^a q^b)$, where $p$ and $q$ are primes. This is achieved by combining:
  • Work of Lemos [39, 40] on the structure of Galois images for non-surjective primes.
  • Ramification results of Smith [58] (Lemma 3) to bound the degree of points when the image is a non-split Cartan subgroup.
  • A new lower bound on the degree of points (Corollary 1) which improves upon previous work by the author and Genao [11].
    The proof demonstrates that for sufficiently large primes, the degree of any point on $X_1(p^a q^b)$ with a rational $j$-invariant exceeds the genus of the curve, rendering the point non-isolated.
• Sharpened Torsion Bounds (Theorem 4): For non-CM elliptic curves $E/F$ with $j(E) \in \mathbb{Q}$, the paper proves that for any $\epsilon > 0$, there exists a constant $C_\epsilon$ such that:
  $$\exp E(F)_{\text{tors}} \leq C_\epsilon \cdot [F:\mathbb{Q}]^{1/2 + \epsilon}$$
  $$\#E(F)_{\text{tors}} \leq C_\epsilon \cdot [F:\mathbb{Q}]^{1 + \epsilon}$$
  This improves the exponent of the degree bound by a square root factor compared to previous results [11, 18]. The exponent $1/2$ for the exponent of the torsion group is shown to be nearly optimal.
• Experimental Data and Classification: The paper provides a comprehensive table of known non-CM isolated $j$-invariants of degree $\leq 10$, justifying their isolation via Jacobian rank computations and gonality bounds.

Significance and Claims
The paper claims that Question 1 serves as a critical "refinement" of the broader uniformity problems in the field. By establishing that the finiteness of isolated $j$-invariants implies strong bounds on isogenies and torsion, the work suggests that attacking the finiteness of these specific points is a viable pathway to resolving broader conjectures.

The results for rational $j$-invariants represent a significant step toward a full classification, as they eliminate the possibility of infinite families of isolated $j$-invariants for curves with levels of the form $p^a q^b$. The author notes that while the current control on "entanglement" (the interaction between torsion fields of distinct primes) is insufficient to prove Hypothesis 2 unconditionally for all rational $j$-invariants, experimental data suggests the problem may be accessible via formal immersion arguments, particularly because the relevant "entanglement modular curves" often possess nontrivial rank 0 quotients, unlike the fiber products of standard modular curves.

The paper concludes that while the full finiteness of isolated $j$-invariants remains open, the methods developed here successfully sharpen the known bounds on torsion for rational $j$-invariants and clarify the logical landscape connecting these uniformity problems.