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

Abstract: Characterizing isolated points on the modular curve $X_1(N)$ is a key obstruction to classifying all points of a fixed degree. These points do not lie in infinite parameterized families, making them difficult to obtain through geometric constructions. In this paper, we focus on the collection of "isolated $j$-invariants" for $X_1(N)$, which are the values obtained by mapping isolated points to the $j$-line. Prior work of the author in collaboration with Ejder, Liu, Odumodu, and Viray asks whether there are only finitely many isolated $j$-invariants lying in extensions of bounded degree. Here, we explore how this question relates to other uniformity problems in the field and give new finiteness results for isolated $j$-invariants in $\mathbb{Q}$. As an application, we show similar methods give sharpened polynomial bounds on torsion for non-CM elliptic curves having rational $j$-invariant.

• The paper proves the finiteness of rational isolated $j$-invariants for modular curves $X_1(p^a q^b)$ where $p, q$ are primes, using a structural theorem and specific cases with detailed ramification and torsion argument details, notably a prime case left incomplete as establishing.
• The research establishes a sharp improvement in polynomial torsion bounds for non-CM elliptic curves with rational $j$-invariants, showing an exponent of $1/2 + extit{$\epsilon$}$ in the degree and concluding that no CM curve bounds exceeded the absolute minimum bound can uniformly hold unless adjusting for CM curves.
• The finiteness of isolated $j$-invariants is linked to generalized Serre's Uniformity Conjecture, and the paper analyzed the implications and interdependencies among four hypotheses (H1-H4), deriving statement implications and constructive bounds

Overview and motivation

This paper by Abbey Bourdon studies "isolated $j$-invariants" attached to the modular curves $X_1(N)$ (2608.19354). A closed point $x \in X_1(N)$ is isolated if it belongs to no infinite parameterized family of points of the same degree — equivalently, it is neither $\mathbb{P}^1$-parameterized nor AV-parameterized via the Abel–Jacobi map $\mathrm{Sym}^d(C) \to \mathrm{Jac}(C)$. Isolated points are precisely the obstruction to extending the classification of degree-$d$ points on $X_1(N)$ to degrees $5 \le d \le 9$, as established by Derickx–van Hoeij and Najman–Varivoda. Mapping an isolated point to the $j$-line yields an isolated $j$-invariant, and the central question (posed previously by Bourdon, Ejder, Liu, Odumodu, and Viray) asks whether there are only finitely many such invariants of each fixed degree. While Faltings' theorem guarantees finitely many isolated points on any fixed curve of fixed degree, this does not imply uniformity across all levels $X_1(N)$0, since the degree of the image on $X_1(N)$1 may drop. The paper makes two contributions: a structural theorem relating the finiteness question to other uniformity conjectures, and new unconditional finiteness and torsion-bounding results for non-CM elliptic curves with rational $X_1(N)$2-invariant.

Implications among uniformity hypotheses

The paper formulates four hypotheses for each degree $X_1(N)$3: (H1) a generalization of Serre's Uniformity Conjecture asserting that mod-$X_1(N)$4 images contain $X_1(N)$5 for large $X_1(N)$6; (H2) finiteness of isolated $X_1(N)$7-invariants of degree $X_1(N)$8; (H3) absence of non-cuspidal non-CM degree-$X_1(N)$9 points on $x \in X_1(N)$0 for large $x \in X_1(N)$1; and (H4) refined polynomial bounds on $x \in X_1(N)$2 and $x \in X_1(N)$3. The main structural result is that H1 implies H2, and H2 implies both H3 and H4.

The implication H2 ⟹ H3 proceeds by lifting a degree-$x \in X_1(N)$4 point on $x \in X_1(N)$5 to $x \in X_1(N)$6 with degree at most $x \in X_1(N)$7, which falls below half the $x \in X_1(N)$8-gonality for large $x \in X_1(N)$9 by Abramovich's gonality bounds; Frey's criterion then renders the lift sporadic, hence isolated, so its $\mathbb{P}^1$0-invariant lies in a finite list, and Serre's Open Image Theorem bounds the possible levels. The implication H2 ⟹ H4 uses a key technical lemma: assuming H2, every point $\mathbb{P}^1$1 associated to a curve with $\mathbb{P}^1$2 satisfies $\mathbb{P}^1$3, combining a uniform-level argument (building on work of Circu-Theodorou and BELOV) with Abramovich gonality bounds. Consequently, a point of order $\mathbb{P}^1$4 over a field of degree $\mathbb{P}^1$5 forces $\mathbb{P}^1$6. Notably, the paper observes that the proof of the related claim in Clark–Pollack relies on a result of Lombardo–Radicke containing an error, so the present argument repairs that gap. The author also notes that if Hypothesis 2 is assumed only for prime levels $\mathbb{P}^1$7, H3 still follows — a useful weakening.

Finiteness for $\mathbb{P}^1$8

The first unconditional main result states that there are only finitely many rational isolated $\mathbb{P}^1$9-invariants arising from modular curves $\mathrm{Sym}^d(C) \to \mathrm{Jac}(C)$0 with $\mathrm{Sym}^d(C) \to \mathrm{Jac}(C)$1 prime. The proof handles three cases. If both primes satisfy $\mathrm{Sym}^d(C) \to \mathrm{Jac}(C)$2, uniform boundedness of $\mathrm{Sym}^d(C) \to \mathrm{Jac}(C)$3-adic image levels reduces to finitely many target curves, each having finitely many isolated points by Faltings. If $\mathrm{Sym}^d(C) \to \mathrm{Jac}(C)$4 with surjective mod-$\mathrm{Sym}^d(C) \to \mathrm{Jac}(C)$5 image, the fiber product structure theorem of BELOV shows the point projects isometrically onto an isolated point of $\mathrm{Sym}^d(C) \to \mathrm{Jac}(C)$6, which was classified in prior joint work. The difficult case is $\mathrm{Sym}^d(C) \to \mathrm{Jac}(C)$7 with $\mathrm{Sym}^d(C) \to \mathrm{Jac}(C)$8. Here the paper establishes a sharpened ramification-theoretic input: building on Smith's work on local Galois representations, if $\mathrm{Sym}^d(C) \to \mathrm{Jac}(C)$9 is the field cut out by torsion coprime to $d$0, then adjoining a point of order $d$1 requires degree at least $d$2 locally. This improves the factor appearing in Bourdon–Genao's Proposition 5 from a loss depending on potentially bad reduction to the absolute constant $d$3, using the fact that for $d$4 the minimal extension attaining good reduction has degree dividing 12 (indeed in $d$5 after accounting for the relevant cases). Combined with Lemos' analysis of when the non-surjective prime divisors interact, the resulting lower bound $d$6 exceeds the genus of $d$7, ruling out isolation. The exceptional 2-adic images 4.8.0.2, 4.16.0.2, and 8.16.0.3 require separate treatment but yield the same conclusion.

Sharpened polynomial torsion bounds

The second main result improves prior bounds of Bourdon–Genao and Clark–Pollack by a square root in the degree: for every $d$8 there exists $d$9 such that all non-CM $X_1(N)$0 with $X_1(N)$1 satisfy

$X_1(N)$2

Two features merit emphasis. First, the exponent $X_1(N)$3 is essentially optimal: any elliptic curve over $X_1(N)$4 acquires a point of prime order $X_1(N)$5 over an extension of degree at most $X_1(N)$6, so no bound with exponent below $X_1(N)$7 can hold uniformly. Second, this is stated to be the first exponent bound too small to accommodate CM curves — the CM curve with $X_1(N)$8 achieves $X_1(N)$9 along suitable fields, so CM exclusion is genuinely necessary rather than an artifact of the method.

The proof combines the lower-bound proposition $5 \le d \le 9$0 with Robin's bound $5 \le d \le 9$1; the contribution of the factor $5 \le d \le 9$2 is absorbed into $5 \le d \le 9$3, yielding $5 \le d \le 9$4. Since $5 \le d \le 9$5 divides $5 \le d \le 9$6, the full-torsion bound follows.

Relation to Hindry–Silverman and Serre uniformity

A final section proves a conditional converse: if there exists an absolute constant $5 \le d \le 9$7 with $5 \le d \le 9$8 for all non-CM $5 \le d \le 9$9 with $j$0, then generalized Serre uniformity (Hypothesis 1) holds. The argument runs the standard dichotomy for small image — Borel, split Cartan normalizer, or non-split Cartan normalizer, excluding exceptional projective images $j$1 via Ghate–Parent for $j$2 — and in each case constructs a field of controlled degree containing a point of order $j$3 or $j$4, contradicting the hypothetical bound. This connects the torsion-bounding question to a problem of Hindry and Silverman and to Breuer's construction showing $j$5 is asymptotically sharp for the full torsion subgroup.

Examples and computational justification

The paper tabulates known non-CM isolated $j$6-invariants of degree up to 10, drawing primarily on van Hoeij's tables. There are exactly four rational ones, conjectured (elsewhere) to be complete, arising from $j$7, $j$8, and $j$9; higher-degree examples include 15 non-CM isolated $j$0-invariants of degree 9. Justification that these points are isolated uses rank-zero hypotheses on Jacobians (via the classification of degree-4 points), known gonalities for $j$1, and the genus/gonality bound for $j$2; non-CM status follows from a degree lower bound for CM points of degree greater than one.

Limitations and open questions

Several restrictions are conceded explicitly. The finiteness theorem covers only products of two distinct primes; the corollary's loss factor of $j$3 prevents extension to three or more primes, since the resulting degree bounds fall short of the genus. Removing the factor $j$4 amounts to controlling horizontal Galois entanglement between $j$5 and $j$6 when $j$7; the paper identifies specific LMFDB curves (e.g., 15.60.3.f.1, 21.126.4.a.1) where the factor cannot currently be removed, though notably each possesses a nontrivial rank 0 quotient unlike the naive fiber product $j$8 — suggesting formal immersion arguments may eventually apply. Hypothesis 2 itself remains unproven even for rational $j$9-invariants. Finally, the torsion bound requires $X_1(N)$00 slack and applies only to non-CM curves, and the equivalence-type statement connecting absolute torsion constants to Serre uniformity is conditional on the Hindry–Silverman-style bound holding.

Conclusion

This paper positions the finiteness of isolated $X_1(N)$01-invariants as an intermediate uniformity statement, strictly weaker than generalized Serre uniformity yet strong enough to imply both non-CM isogeny bounds for $X_1(N)$02 and refined polynomial torsion bounds. Unconditionally, it settles the question for levels supported on at most two primes over $X_1(N)$03 and delivers the sharpest known torsion exponents ($X_1(N)$04 on the exponent, $X_1(N)$05 on cardinality) for non-CM curves with rational $X_1(N)$06-invariant, while demonstrating via the conditional converse that sufficiently strong absolute torsion bounds would recover Serre uniformity itself. The remaining obstructions are localized precisely in entanglement phenomena, for which the paper provides concrete computational targets in the LMFDB.