2025/05/22 by Luís Dieulefait, Dieulefait, Luís, Josep González +3
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2505.16529
openalex publication_date 2025/05/22 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
We say that a normalized modular form is of CM type modulo ℓ by an imaginary quadratic field K if its Fourier coefficients ap are congruent to 0 modulo a prime \mathcal L| ℓ for every prime p that is inert in K. In this paper, we address the following question. Let f be a weight~2 cuspidal Hecke eigenform without complex multiplication which is of CM type modulo ℓ by an imaginary quadratic field K. Does there exist a congruence modulo ℓ between f and a genuine CM modular form of weight~2? We conjecture that such a congruence always exists. We prove this conjecture for ℓ>2 and ℓ≠ 3 when K=ℚ(√(-3)). In this setting, we discuss three situations: (i) modular forms attached to abelian surfaces with quaternionic multiplication, (ii) ℚ-curves completely defined over an imaginary quadratic field, and (iii) elliptic curves over ℚ whose 5-torsion Galois representation has image the maximal cyclic of order 16 inside GL2(\mathbb F5). In all these cases, the modular forms under consideration are of CM type modulo suitable primes~ℓ, and we show that the associated residual Galois representations are monomial with respect to an imaginary quadratic field K (in some instances, more than one such field). Finally, we present numerical evidence that motivated the conjecture and provides further support for its validity beyond the cases treated in this paper.