Finiteness of isolated j-invariants in each fixed degree

Determine whether there are only finitely many isolated j-invariants of each fixed degree over the rational numbers.

Background

The paper studies isolated points on the modular curves X_1(N) and their images under the natural map to the j-line. Although each individual modular curve has only finitely many isolated points of any fixed degree, this does not immediately imply that the associated isolated j-invariants remain finite as N varies. The unresolved question asks whether the degree of the j-invariant alone imposes a global finiteness bound across all levels N.

The paper proves several implications relating this finiteness question to uniformity conjectures for Galois representations, isogenies, and torsion, and establishes unconditional finiteness for rational isolated j-invariants arising from curves X_1(pa qb).

References

Are there only finitely many isolated $j$-invariants of each fixed degree?

— On the Finiteness of Isolated $j$-invariants for $X_1(N)$  (2608.19354 - Bourdon, 19 Aug 2026) in Question 1.1, Section 1

The 4 non-CM isolated $j$-invariants in $Q$ are conjectured to be the only ones .

— On the Finiteness of Isolated $j$-invariants for $X_1(N)$  (2608.19354 - Bourdon, 19 Aug 2026) in Section 1, paragraph following Table 1

This leaves open the possibility that one could prove there are only finitely many isolated $j$-invariants in $Q$ using formal immersion arguments, as in other uniformity results.

— On the Finiteness of Isolated $j$-invariants for $X_1(N)$  (2608.19354 - Bourdon, 19 Aug 2026) in Remark 4.6, Section 4, subsection “Proof of Theorem \ref{finiteness_2_primes}”