Local-to-global characterization of prime-detecting quasimodular forms

Determine whether there exists, for each even integer k ≥ 6, an explicit infinite list of tuples (A_j, B_j, M_j) such that a mixed-weight quasimodular form f in \widetilde{M}_{\leq k} with integral Fourier coefficients belongs to the prime-detecting submodule \Omega if and only if its coefficients satisfy the Ramanujan-type congruences a_f(A_j n+B_j) ≡ 0 (mod M_j) for every j and all n ≥ 0.

Background

The paper observes that universal Ramanujan-type congruences satisfied by all prime-detecting quasimodular forms provide necessary tests for membership in the submodule \Omega. For a fixed progression and modulus, the forms satisfying the corresponding coefficient congruence form the kernel of a homomorphism from a finite-dimensional space of mixed-weight quasimodular forms to a higher-level, character-twisted coefficient space.

The unresolved issue is whether an explicit infinite collection of such local congruence conditions can characterize \Omega exactly, rather than merely provide necessary conditions. The authors note that no finite collection of moduli can settle this characterization and leave open whether a local-to-global argument can establish an affirmative answer.

References

Given k \geq 6 even and f \in \widetilde{M}_{\leq k} \cap [[q]], is there an explicit (infinite) list of tuples $(A_j, B_j, M_j)$ such that $f$ obeys the Ramanujan-type congruences $a_f(A_j n + B_j) \equiv 0 \pmod{M_j}$ if and only if $f \in \Omega$? No finite collection of moduli could resolve such a question, but Theorem \ref{Thm: Infinitely many uniform congruences} leaves open whether such a question can be answered affirmatively using some local-to-global argument.

— Universal Ramanujan-type congruences for prime-detecting quasimodular forms  (2608.18892 - Chrobak et al., 19 Aug 2026) in Section Conclusion and Conjectures, subsection “Conjectural classification of \Omega by congruences”