On the modularity level of modular abelian varieties over number fields

Abstract

Let $f$ be a weight two newform for $\Gamma_1(N)$ without complex multiplication. In this article we study the conductor of the absolutely simple factors $B$ of the variety $A_f$ over certain number fields $L$. The strategy we follow is to compute the restriction of scalars $\operatorname{Res}_{L / Q}(B)$, and then to apply Milne's formula for the conductor of the restriction of scalars. In this way we obtain an expression for the local exponents of the conductor $\mathcal{N}_L(B)$. Under some hypothesis it is possible to give global formulas relating this conductor with $N$. For instance, if $N$ is squarefree we find that $\mathcal{N}_L(B)$ belongs to $\mathbb{Z}$ and $\mathcal{N}_L(B) \mathfrak{f}_L^{\operatorname{dim} B}=N^{\operatorname{dim} B}$, where $\mathfrak{f}_L$ is the conductor of $L$.