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

TL;DR

This work determines the Hecke fields K_f for weight one exotic newforms of A_4-, S_4-, and A_5-type, expressing K_f explicitly in terms of the nebentypus order d. It offers a complete classification of the Galois image ρ_f(G_Q) and provides refined descriptions in the S_4-case via a comparison between χ^{d/2} and sgn ∘ ρ̄_f. The authors introduce twist-minimal and strongly minimal newforms, derive restrictions on the possible nebentypus orders, and construct families realizing these orders. They also study densities of primes generating K_f through a Chebotarev-based framework and describe generators of K_f by eigenvalues a_p(f), linking the arithmetic of nebentypus and projective images to explicit field generation results.

Abstract

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.

Theorems & Definitions (142)

• Remark 1.1
• Theorem 1.2: Theorem \ref{['thm:mainA4general']}
• Theorem 1.3: Theorem \ref{['thm:mainA5general']}
• Theorem 1.4: Theorem \ref{['thm:mainS4general']}
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Theorem 1.8: Theorem \ref{['thm:S4_refinement']}
• Remark 1.9
• Theorem 1.10: Theorem \ref{['thm:density_A_4_case']}