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PreprintCC BY 4.0arXiv

On the saturability of p-adic Lie groups

  • Casey Perdue
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Abstract

Lazard theory is becoming increasingly important in p-adic representation theory, especially when studying mod p representations. For G a connected reductive group defined over a finite extension K/ℚp, a question going back to Lazard is when a pro-p Iwahori subgroup of G(K) admits a saturated p-valuation. In this article, we establish a deep connection to Moy-Prasad theory, and show that the Moy-Prasad filtration defines a saturated p-valuation on Gx,1/(p−1)+ for any point x in the semisimple Bruhat-Tits building of G, given a mild assumption on the maximal torus of G and that p>2. We also compute Lazard Lie algebras defined over ℤp and the polynomial ring 𝔽p[ε] associated to the saturated group Gx,1/(p−1)+, which allows us to reinterpret the Moy-Prasad isomorphism as an isomorphism of Lie algebras over 𝔽p[ε].

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Cite

BibTeXRIS
BibTeX
@misc{perdue2026saturability,
  title = {{On the saturability of $p$-adic Lie groups}},
  author = {Casey Perdue},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13877},
  eprint = {2610.13877},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13877},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the saturability of p-adic Lie groups
AU  - Casey Perdue
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13877
UR  - https://arxiv.org/abs/2610.13877
AB  - Lazard theory is becoming increasingly important in p-adic representation theory, especially when studying mod p representations. For G a connected reductive group defined over a finite extension K/ℚ_p, a question going back to Lazard is when a pro-p Iwahori subgroup of G(K) admits a saturated p-valuation. In this article, we establish a deep connection to Moy-Prasad theory, and show that the Moy-Prasad filtration defines a saturated p-valuation on G_x,1/(p−1)+ for any point x in the semisimple Bruhat-Tits building of G, given a mild assumption on the maximal torus of G and that p>2. We also compute Lazard Lie algebras defined over ℤ_p and the polynomial ring 𝔽_p[ε] associated to the saturated group G_x,1/(p−1)+, which allows us to reinterpret the Moy-Prasad isomorphism as an isomorphism of Lie algebras over 𝔽_p[ε].
ER  -