Abelian varieties, \(\ell\)-adic representations, and \(\ell\)-independence
Let \(G=\text{Gal} (\overline{K}/K)\) be the Galois group of the extension of a global field \(K\) by a separable closure \(\overline{K}\). For any abelian variety \(A\) of dimension \(g\) over \(K\), the Galois group \(G\) acts on the group \(A[\ell^n]\) of \(\ell^n\)-torsion points of \(A\) where \(\ell\) is a prime number different from \(\text{char} (K)\). Therefore, \(G\) acts continuously on the vector space \(V_\ell=(\displaystyle\varprojlim_n A[\ell^n]) \otimes_{\mathbb{Z}_\ell}, \mathbb{Q}_\ell\cong \mathbb{Q}_\ell^{2g}\), and the corresponding system \(\rho_\ell: \Aut(V_\ell)\), \(\ell\neq\text{char} (K)\), of representations is strictly compatible in the sense of Serre. Let \(G_\ell\) denote the image of \(G\) under \(\rho_\ell\) in \(\text{GL} (2g,\mathbb{Q}_\ell)\), \(G_\ell^0\) the identity component of \(G_\ell\), and \(\rho_\ell^{\text{alg}}\) the induced representation of \(G_\ell^0\) and \(V_\ell\). The present paper is devoted to a closer study of the identity component \(G_\ell^0\), and that mainly under the aspect of searching for more evidence for the following well-known conjecture: There exists a connected reductive group \(H\) over \(\mathbb{Q}\), together with a faithful representation \(\rho\) of \(H\) on a \(\mathbb{Q}\)-vector space \(V\) auch that for sufficiently big prime numbers \(\ell\neq \text{char}(K)\) there is an isomorphism of representations \((G_\ell^0, V_\ell, \rho_\ell^{\text{alg}})= (H,V,\rho) \otimes_\mathbb{Q} \mathbb{Q}_\ell\). Building upon the results obtained in two earlier papers [cf. \textit{M. Larsen} and \textit{R. Pink}, Invent. Math. 102, No. 2, 377-398 (1990; Zbl 0687.22004) and ibid. 107, No. 3, 603-636 (1992; Zbl 0778.11036)] the authors prove that \(G_\ell^0\) is unramified for \(\ell\gg 0\), and that the conjecture is true, at least for primes \(\ell\) that form a set of Dirichlet density one, under additional assumptions on \(\text{End}_{\overline{K}} (A)\), \(g=\dim A\) and the splitting field of the representation system \(\{\rho_\ell\}\). In the special case of a function field \(K\) as ground-field, it is shown that the conjecture has an affirmative answer over the complex numbers with respect to all embeddings of \(\mathbb{Q}_\ell\) in \(\mathbb{C}\). Finally, the subjeet is discussed in the context of the famous Mumford-Tate conjecture for abelian varieties over number fields. More precisely, another main result of the present work implies that the Mumford-Tate conjecture holds for every prime if and only if it holds for at least one prime \(\ell\). In a concluding section, the authors discuss the interrelation between the aforementioned conjecture on \(\ell\)-adic representations, the general Tate conjecture, and the semisimplicity conjecture.
scientific article; zbMATH DE number 799560