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INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE
Published online by Cambridge University Press: 07 June 2018
Abstract
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.
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- Research Article
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- Copyright
- © Cambridge University Press 2018
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