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The trianguline variety for reductive groups

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Andrea Conti, Mohamed Moakher, Julian Quast

Number Theory · Submitted 2 Jul 2026 · 21 references

Pith's verdict: Straightforward extension of the Breuil-Hellmann-Schraen local structure theorem to split connected reductive groups, plus a crystallinity criterion.

T0 · AI review

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arXiv ID2607.02215 (version 1)
Pith IDWHOIIVIZ, Pith's permanent record of this paper and its review
CitationAndrea Conti, Julian Quast, Mohamed Moakher. "The trianguline variety for reductive groups." arXiv:2607.02215 (2026). https://pith.science/paper/WHOIIVIZ
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The paper

The paper claims

For split connected reductive groups the trianguline variety is smooth over loci set by regularity conditions on the triangulation parameter and normal at certain points outside those loci.

The reading

This paper studies the trianguline variety attached to split connected reductive groups. It extends an existing theorem on local structure by proving smoothness of the variety precisely over the loci where the triangulation parameter meets various regularity conditions. It also establishes normality at selected points that lie outside those smooth loci. A separate crystallinity criterion is shown for (φ, Γ_K)-modules carrying G-structure. These local properties describe the geometry of a space that encodes structured Galois representations.

What carries the argument

The trianguline variety for split connected reductive groups, whose local smoothness and normality are controlled by regularity conditions on the triangulation parameter.

Extended reading notes

Core claim

The trianguline variety for split connected reductive groups is smooth over the loci determined by various regularity conditions on the triangulation parameter, and is normal at certain points outside of these smooth loci. Along the way a crystallinity criterion is proved for (φ, Γ_K)-modules with G-structure.

Load-bearing premise

The groups under study are split and connected reductive groups.

Editorial extensions

If this is right

  • Smoothness holds over all loci fixed by the listed regularity conditions on the triangulation parameter.
  • Normality is obtained at the indicated points lying outside the smooth loci.
  • The crystallinity criterion applies directly to (φ, Γ_K)-modules equipped with G-structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The local smoothness results may simplify the computation of irreducible components or dimensions of the variety in concrete cases.
  • The crystallinity criterion could be tested on explicit filtered phi-modules with group actions arising from known Galois representations.
  • Methods used here might be adapted to study analogous local properties for trianguline varieties attached to groups that are not split.
Original abstract

We study the trianguline variety for split connected reductive groups. We generalize a theorem of Breuil, Hellmann, and Schraen about its local structure, establishing smoothness over the loci determined by various regularity conditions on the triangulation parameter, and normality at certain points outside of these smooth loci. Along the way, we prove a crystallinity criterion for $(\varphi,\Gamma_K)$-modules with $\mathsf G$-structure.

References

Discussion

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