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arXiv · 2303.06549

On the observability of Galois representations and the Tate conjecture

Yves André
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Abstract

The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using $p$-adic Hodge theory. We show that an unconditional result lies behind this implication: the {\it observability} of arithmetic monodromy groups of geometric origin (in any characteristic) - which leads to a sharpening of Moonen's result. We also discuss another aspect of the Tate conjecture related to the transcendence of $p$-adic periods.

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Yves André. 2023-03-12. On the observability of Galois representations and the Tate conjecture. https://arxiv.org/abs/2303.06549

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