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Peer-reviewedCC BY 4.0Journal of Number Theory

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 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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Cite

BibTeXRIS
BibTeX
@article{andre2026observability,
  title = {{On the observability of Galois representations and the Tate conjecture}},
  author = {Yves André},
  journal = {Journal of Number Theory},
  year = {2026},
  month = oct,
  doi = {10.1016/j.jnt.2023.07.010},
  eprint = {2303.06549},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05844588v1},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - On the observability of Galois representations and the Tate conjecture
AU  - Yves André
PY  - 2026
DA  - 2026/10/29
JO  - Journal of Number Theory
DO  - 10.1016/j.jnt.2023.07.010
UR  - https://hal.science/hal-05844588v1
AB  - 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 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.
SN  - 0022-314X
SN  - 1096-1658
ER  -