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  • Anna Cadoret (a1) and Ben Moonen (a2)

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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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
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