On the modularity level of modular abelian varieties over number fields

TL;DR

This paper investigates the conductor properties of simple factors of modular abelian varieties over number fields, providing formulas relating their conductors to the original level and field conductors under certain conditions.

Contribution

It introduces a method to compute the conductor of simple factors of modular abelian varieties over number fields using restriction of scalars and Milne's formula, with explicit global relations.

Findings

Derived formulas for local exponents of conductors

Established conditions under which conductors relate to original levels and field conductors

Showed that for squarefree levels, conductors are integers and relate multiplicatively to field conductors

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 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 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 N_L(B) belongs to Z and N_L(B)*f_L^{dim B}=N^{dim B}, where f_L is the conductor of L.