We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Let 
$Y$ be an abelian variety over a subfield 
$k\subset \mathbb{C}$ that is of finite type over 
$\mathbb{Q}$. We prove that if the Mumford–Tate conjecture for 
$Y$ is true, then also some refined integral and adelic conjectures due to Serre are true for 
$Y$. In particular, if a certain Hodge-maximality condition is satisfied, we obtain an adelic open image theorem for the Galois representation on the (full) Tate module of 
$Y$. We also obtain an (unconditional) adelic open image theorem for K3 surfaces. These results are special cases of a more general statement for the image of a natural adelic representation of the fundamental group of a Shimura variety.
