Classifications of Hecke Fields and Galois Images of Weight One Exotic Newforms

We determine the Hecke fields associated with weight one newforms of $A_4$-, $S_4$-, and $A_5$-type, expressed in terms of the order of its nebentypus. Furthermore, for each type, we provide a complete classification of the images of the corresponding Galois representations.

💡 Research Summary

The paper studies weight‑one newforms whose projective Galois image is one of the non‑dihedral finite groups A₄, S₄ or A₅, and provides a complete description of both their Hecke fields and the full Galois image.

Main objects.
Let f be a weight‑one newform of level N with nebentypus χ (odd, so its order d is even). By Deligne–Serre, f gives an odd, continuous, irreducible 2‑dimensional complex Galois representation ρ_f : G_ℚ → GL₂(ℂ). Its projective image (\bar\rho_f) lies in PGL₂(ℂ) and is isomorphic to D_{2n}, A₄, S₄ or A₅. The paper focuses on the exotic cases A₄, S₄, A₅.

Hecke field K_f.
Define K_f = ℚ({a_n(f)}), the field generated by all Fourier coefficients. The authors first partition the set of primes p (coprime to N) into subsets R_m according to the order m of (\bar\rho_f(\mathrm{Frob}_p)). Lemma 3.2 shows that a_p(f) generates a field of the form ℚ(p χ(p)), ℚ(√5, p χ(p)) etc., depending on m.

• A₄‑type: Theorem 3.3 proves K_f = ℚ(ζ_{2d}). The proof uses Chebotarev density to show that the set of primes where (\bar\rho_f(\mathrm{Frob}p)) has order 1 or 3 has density 1/12, which forces the inclusion ℚ(ζ{2d}) ⊂ K_f.

• A₅‑type: Theorem 3.5 gives K_f = ℚ(ζ_{2d}, √5). The extra √5 appears because the projective image contains elements of order 5, leading to a quadratic subfield.

• S₄‑type: Theorem 3.7 distinguishes three cases according to the 2‑adic valuation ord₂(d).

  • If ord₂(d)=1, K_f is either ℚ(ζ_d, √{-2}) or ℚ(ζ_{4d}).
  • If ord₂(d)=2, K_f is either ℚ(ζ_d) or ℚ(ζ_{2d}).
  • If ord₂(d)≥3, K_f = ℚ(ζ_{2d}).
    Theorem 3.9 supplies a precise criterion: compare χ^{d/2} with the sign character of (\bar\rho_f). Equality selects one of the two possibilities in each ambiguous case.

Generation by a single coefficient.
For each type the authors define P_f = {p | gcd(p,N)=1, K_f = ℚ(a_p(f))} and compute its Dirichlet density.

• A₄‑type (Theorem 4.1): density is 3φ(d)/(4d) when ker(χ) is not contained in the kernel of the abelianization map A₄ → A₄^{ab}, otherwise φ(d)/d.
• A₅‑type (Theorem 4.2): density = 2φ(d)/(5d) if 5∤d, and 3φ(d)/(4d) if 5|d.
• S₄‑type (Theorem 4.3): several sub‑cases depending on ord₂(d) and the relation between χ^{d/2} and sgn ∘ (\bar\rho_f). In particular, when ord₂(d)=1 and χ^{d/2}≠sgn ∘ (\bar\rho_f), P_f is empty, i.e. no single coefficient generates K_f.

Strongly minimal newforms.
The paper introduces “strongly minimal” newforms: they are twist‑minimal and have minimal ramification at every prime among all twist‑minimal forms in the same twist class. Theorem 5.3 classifies the possible orders d of the nebentypus for such forms:

• A₄‑type: d = 3·2^k (k≥1).
• A₅‑type: d = 2^k·3^{δ₃}·5^{δ₅} with δ₃,δ₅∈{0,1}.
• S₄‑type: d = 2^k or d = 3·2^k (k≥1).
  Propositions 5.14, 5.21, 5.22 construct explicit strongly minimal newforms for every admissible d, confirming that the bounds are sharp.

Full Galois image.
The final section determines the exact group structure of ρ_f(G_ℚ).

• A₄‑type (Theorem 6.7, Prop 6.10):

  • If ker(χ) is not contained in the kernel of the abelianization A₄ → A₄^{ab}, then
    (\rho_f(G_ℚ) ≅ (SL₂(𝔽₃) × ℤ/2dℤ)/⟨(-I,d)⟩.)
  • If the inclusion holds, then d is divisible by 3 and
    (\rho_f(G_ℚ) ≅ (SL₂(𝔽₃) × ℤ/3ℤ × ℤ/2dℤ)/⟨(-I,d)⟩,) where the ℤ/3ℤ factor comes from the abelianization of SL₂(𝔽₃).

• A₅‑type (Theorem 6.8):
  (\rho_f(G_ℚ) ≅ (SL₂(𝔽₅) × ℤ/2dℤ)/⟨(-I,d)⟩.)

• S₄‑type (Theorem 6.9, Prop 6.14): Let BO₄₈ be the binary octahedral group (order 48) with central element z of order 2.

  • If χ^{d/2} ≠ sgn ∘ (\bar\rho_f), then
    (\rho_f(G_ℚ) ≅ (BO₄₈ × ℤ/2dℤ)/⟨(z,d)⟩.)
  • If χ^{d/2} = sgn ∘ (\bar\rho_f), then
    (\rho_f(G_ℚ) ≅ (BO₄₈ × ℤ/2ℤ × ℤ/2dℤ)/⟨(z,d)⟩.)

These descriptions exhibit the Hecke field as the cyclotomic part ℤ/2dℤ (or ℤ/2dℤ together with a small extra factor) and the non‑abelian simple part as SL₂ over a small finite field or the binary octahedral group, depending on the projective type.

Broader context and applications.
The authors note that Miyazawa’s forthcoming work on dihedral weight‑one newforms, combined with their results, implies that for any fixed integer n only finitely many number fields of degree ≤ n arise as Hecke fields of weight‑one newforms. Moreover, the density calculations provide a weight‑one analogue of results by Ko–Stein–Wiese on generation of Hecke algebras by a single eigenvalue for higher weight forms.

Conclusion.
The paper achieves a complete, explicit classification of both the Hecke fields and the full Galois images attached to weight‑one exotic newforms. By expressing everything in terms of the nebentypus order d and by introducing the notion of strongly minimal newforms, the authors not only settle structural questions but also give constructive methods to produce examples for every admissible d. This work deepens the understanding of the interplay between modular forms, cyclotomic extensions, and finite subgroups of PGL₂(ℂ), and it opens the way for further arithmetic investigations of weight‑one forms and their associated Galois representations.