Finiteness of strong Hecke-eigenform congruence classes

Establish, for every fixed level N, prime p not dividing N, and integer m at least 1, that only finitely many modulo p^m congruence classes of strong Hecke eigenforms on Γ₁(N) occur as the weight varies.

Background

The paper studies congruences between systems of Hecke eigenvalues of characteristic-zero normalized eigenforms as the weight varies. The conjecture asks for finiteness of these congruence classes at fixed level, prime, and prime-power modulus.

An equivalent formulation is the existence of a strong weight bound B(N,p,m) such that every normalized characteristic-zero eigenform on Γ₁(N) is congruent modulo pm to an eigenform of weight at most B(N,p,m). The paper proves several level-one cases but does not resolve the conjecture for general N, p, and m.

References

Questions of this kind belong to the general study of modular forms and Galois representations modulo prime powers. For fixed level N, prime p∤N, and integer m≥1, the author, in joint work with Kiming and Wiese, formulated the following finiteness conjecture Conjecture~1. There are only finitely many modulo pm congruence classes of strong Hecke eigenforms on Γ₁(N).

— Prime-power congruences for level-one eigenforms  (2609.16750 - Rustom, 15 Sep 2026) in Section 1, Introduction and main results, subsection “The problem”; Conjecture 1 (label conj:krw-finiteness)