$\mu$-type subgroups of $J_1(N)$ and application to cyclotomic fields

Abstract

Let $p$ be an odd prime number, and $N$ a positive integer prime to $p$. We prove that $\mu$-type subgroups of the modular Jacobian variety $J_1(N)$ or $J_1(Np)$ of order a power of $p$ and defined over some abelian extensions of $\mathbb{Q}$ 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.