On the modularity level of modular abelian varieties over number fields. (English) Zbl 1233.11068
Summary: Let \(f\) be a weight two newform for \(\varGamma _{1}(N)\) 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 \(\text{Res}_{L/\mathbb Q}(B)\), 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 \(\mathcal N_L(B)\mathfrak f _L^{\dim B}\), where \(\mathfrak f _L\) is the conductor of \(L\).
MSC:
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11G10 | Abelian varieties of dimension \(> 1\) |
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