Explicit surjectivity of Galois representations for abelian surfaces and \(\operatorname{GL}_{2}\)-varieties
Let \(A\) be an absolutely simple abelian variety defined over a number field \(K\), without potential complex multiplication. Suppose that \(\dim A=2\) or that \(A\) is of \(\mathrm{GL}_2\)-type, i.e., \(\mathrm{End}_{\overline{K}}(A)\otimes \mathbb{Q}\) is a totally real number, of degree \(\dim A\) over \(\mathbb{Q}\).NEWLINENEWLINEThe author gives an \textit{explicit} bound \(\ell_0\), depending only on \(A\) and \(K\), such that for each each prime \(\ell>\ell_0\) the image of \(\mathrm{Gal}(\overline{K}/K)\) in \(\mathrm{Aut}(T_\ell(A))\) is as large as possible.
scientific article; zbMATH DE number 6597440
scientific article