S-unit differences between an elliptic-curve j-invariant and singular moduli

Establish whether, for a non-CM elliptic curve E/Q and a singular modulus j of discriminant f²Δ* with f≥1, the difference j_E−j can be an S-unit for finite sets S containing all prime divisors of Δ*, thereby determining whether results of this type are available to remove the assumptions on Δ, j_E, and the Kodaira symbols in the paper’s lower-bound theorem.

Background

The paper’s infinitude theorem for primes with a prescribed Frobenius field requires several hypotheses: Δ must satisfy specified congruence and compositeness conditions, the elliptic curve must have an even integral j-invariant, and the Kodaira symbol at every odd bad prime must be I_0*.

The authors explain that removing these hypotheses would require controlling whether differences between the fixed elliptic-curve j-invariant j_E and singular moduli are S-units. Such a result would connect the Frobenius-field problem to the arithmetic of singular moduli and S-unit equations, but the paper states that the availability of suitable results is currently unknown.

References

To remove the assumptions on $\Delta$, $j_E$, and the Kodaira type in \Cref{thm:lower-bound}, it is necessary to show that for a non-CM elliptic curve E/Q and a singular modulus $j$ of discriminant $f2\Delta*$ (f\in Z_{\geq 1}$), the difference $j_E-j$ is not an $S$-unit for certain finite sets $S$ containing all prime divisors of $\Delta*$. To the best of the author’s knowledge, it is not currently known whether results of this type are available in the literature (see, for example, for related discussions).

— Infinitely many primes with a fixed Frobenius field for an elliptic curve over $\mathbb{Q}$  (2608.17639 - Wang, 18 Aug 2026) in Remark 1 (labelled rem:rm-1), Section 1 (Introduction)

Then, it is conjectured that given any finite set of primes $S$, there are only finitely many CM points $x_i$ that are $S$-integral in $A_1 \setminus { x}$.

— S-integrality for families of ordinary K3 surfaces and algebraicity theorems  (2609.25703 - Jiang et al., 22 Sep 2026) in Section 1, subsection “Main results”