Franco-Asian Summer School on Arithmetic Geometry in Luminy / Ecole d'été franco-asiatique sur la géométrie arithmétique à Luminy

Collection Franco-Asian Summer School on Arithmetic Geometry in Luminy / Ecole d'été franco-asiatique sur la géométrie arithmétique à Luminy

Organizer(s) Abbes, Ahmed ; Mézard, Ariane ; Saito, Takeshi ; Zheng, Weizhe
Date(s) 30/05/2022 - 03/06/2022
linked URL https://conferences.cirm-math.fr/2534.html
00:00:00 / 00:00:00
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Automatic de Rhamness of p-adic local systems and Galois action on the pro-algebraic fundamental group

By Alexander Petrov

Given a $p$-adic local system $L$ on a smooth algebraic variety $X$ over a finite extension $K$ of $Q_{p}$, it is always possible to find a de Rham local system $M$ on $X$ such that the underlying local system $\left.L\right|_{X_{\bar{K}}}$ embeds into $\left.M\right|_{X_{\bar{K}}}$. I will outline the proof that relies on the $p$-adic Riemann-Hilbert correspondence of Diao-Lan-Liu-Zhu. As a consequence, the action of the Galois group $G_{K}$ on the pro-algebraic completion of the étale fundamental group of $X_{\bar{K}}$ is de Rham, in the sense that every finite-dimensional subrepresentation of the ring of regular functions on that group scheme is de Rham. This implies that every finite-dimensional subrepresentation of the ring of regular functions on the pro-algebraic completion of the geometric pi $i_{1}$ of a smooth variety over a number field satisfies the assumptions of the Fontaine-Mazur conjecture. Complementing this result, I will sketch a proof of the fact that every semi-simple representation of $G a l(\bar{Q} / Q)$ arising from geometry is a subquotient of the ring of regular functions on the pro-algebraic completion of the fundamental group of the projective line with 3 punctures.

Information about the video

Citation data

  • DOI 10.24350/CIRM.V.19928203
  • Cite this video Petrov, Alexander (30/05/2022). Automatic de Rhamness of p-adic local systems and Galois action on the pro-algebraic fundamental group. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19928203
  • URL https://dx.doi.org/10.24350/CIRM.V.19928203

Bibliography

  • PETROV, Alexander. Geometrically irreducible $ p $-adic local systems are de Rham up to a twist. arXiv preprint arXiv:2012.13372, 2020. - https://arxiv.org/abs/2012.13372
  • PETROV, Alexander. Universality of the Galois action on the fundamental group of $\mathbb {P}^ 1\setminus{0, 1,\infty} $. arXiv preprint arXiv:2109.09301, 2021. - https://arxiv.org/abs/2109.09301

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