A method to prove that a modular Galois representation has large image

2022/05/27 by Mascot, Nicolas
#11F11 #11F80 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2205.14030

Abstract

Let ρ be a mod ℓ Galois representation attached to a newform f. Explicit methods are sometimes able to determine the image of ρ, or even the number field cut out by ρ, provided that ℓ and the level N of f are small enough; however these methods are not amenable to the case where ℓ or N are large. The purpose of this short note is to establish a sufficient condition for the image of ρ to be large and which remains easy to test for moderately large ℓ and N.

Related

• Explicit determination of images of Galois representations attached to Hilbert modular forms
• Computing modular Galois representations for small ℓ
• Computing modular Galois representations
• A level raising result for modular Galois representations modulo prime powers
• Multiplicities of Galois representations of weight one (with an appendix by Niko Naumann)