On lifting representations and actions on curves of the metacyclic groups C_p^srtimes C_m

Abstract

For a prime p, a pair (s,m)∈ℕ² with m relatively prime to p, a homomorphism χ:C_m→Aut(C_p^s), and an algebraically closed field k of characteristic p, we consider the semidirect product G=C_p^srtimes_χ C_m, the p-Sylow subgroup H≅ C_p^s of G, and a k[G]-module V. Let R be a complete discrete valuation ring of residue field k and mixed characteristic (0,p) that contains a primitive p^s-th root of unity. If χ is injective, we present two necessary and sufficient criteria for lifting V to an R[G]-module Ṽ which is a free R-module: (i) when no extra requirement is made on Ṽ and (ii) when we require Ṽ^H={0}. The criteria correct several results in the literature and we use them to prove that, if χ is injective and G acts faithfully on a connected smooth projective curve X over k, then, under mild hypotheses satisfied if X→ X/G is a Harbater--Katz--Gabber cover, the k[G]-module H⁰(X,Ω_X) has a lift Ṽ to R with Ṽ^H={0}. Also, with B:=R[1/p], we prove the following obstruction when p is odd, m is even, χ is injective, and X/C_p^s≅ℙ¹_k: if no such lift Ṽ exists with the B[H]-module Ṽ⊗_R B defined over ℚ, then the action of G on X does not lift to R.

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BibTeX RIS

BibTeX

@misc{dang2026lifting,
  title = {{On lifting representations and actions on curves of the metacyclic groups $C_p^s\rtimes C_m$}},
  author = {Huy Dang and Adrian Vasiu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.03191},
  eprint = {2609.03191},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03191},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On lifting representations and actions on curves of the metacyclic groups C_p^srtimes C_m
AU  - Huy Dang
AU  - Adrian Vasiu
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.03191
UR  - https://arxiv.org/abs/2609.03191
AB  - For a prime p, a pair (s,m)∈ℕ^2 with m relatively prime to p, a homomorphism χ:C_m→Aut(C_p^s), and an algebraically closed field k of characteristic p, we consider the semidirect product G=C_p^srtimes_χ C_m, the p-Sylow subgroup H≅ C_p^s of G, and a k[G]-module V. Let R be a complete discrete valuation ring of residue field k and mixed characteristic (0,p) that contains a primitive p^s-th root of unity. If χ is injective, we present two necessary and sufficient criteria for lifting V to an R[G]-module Ṽ which is a free R-module: (i) when no extra requirement is made on Ṽ and (ii) when we require Ṽ^H={0}. The criteria correct several results in the literature and we use them to prove that, if χ is injective and G acts faithfully on a connected smooth projective curve X over k, then, under mild hypotheses satisfied if X→ X/G is a Harbater--Katz--Gabber cover, the k[G]-module H^0(X,Ω_X) has a lift Ṽ to R with Ṽ^H={0}. Also, with B:=R[1/p], we prove the following obstruction when p is odd, m is even, χ is injective, and X/C_p^s≅ℙ^1_k: if no such lift Ṽ exists with the B[H]-module Ṽ⊗_R B defined over ℚ, then the action of G on X does not lift to R.
ER  -

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