Abelian varieties over number fields, tame ramification and big Galois image (Q2434608)

The authors show that for every natural number \(n\geq 1\) and every number field \(K\) there exists a constant \(\ell_0\) such that for every prime \(\ell\) such that \(\ell_0 \leq \ell\), there exists a finite extension \(F/K\) unramified at all places over \(\ell\) and an \(n\)-dimensional abelian variety \(A/F\) such that the corresponding representation of \(\mathrm{Gal}(F/K)\) on the \(\ell\)-torsion points is maximal, i.e., the image of this representation is \(\mathrm{Gsp}(A[L](\overline{F}))\) and such that this representation is tamely ramified.