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INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 3 / May 2020
- Published online by Cambridge University Press:
- 07 June 2018, pp. 869-890
- Print publication:
- May 2020
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- Article
- Export citation
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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.
