𝜇-type subgroups of 𝐽₁(𝑁) and application to cyclotomic fields

Abstract

Let 𝑝 be an odd prime number, and 𝑁 a positive integer prime to 𝑝. We prove that 𝜇-type subgroups of the modular Jacobian variety 𝐽₁(𝑁) or 𝐽₁(𝑁𝑝) of order a power of 𝑝 and defined over some abelian extensions of ℚ are trivial, under several hypotheses. For the proof, we use the method of Vatsal. As application, we show that a conjecture of Sharifi is valid in some cases.