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On multilinear forms for mod p representations of GL2 over a p-adic field

  • Hao Liu
  • Haoran Wang
  • Yuting Yang
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Abstract

Let F be a finite extension of ℚp, for any prime p, and let L/F be a ramified quadratic extension. We compute the spaces of 𝒪L×-equivariant linear forms on the representation indGKZσ/(Φ−λ). The only nonzero cases arise from one-dimensional quotients. Consequently, the restriction of an infinite-dimensional irreducible smooth mod p representation of GL2(F) with a central character to 𝒪L× has no character quotient. We apply this vanishing result to show that a finite tensor product of smooth mod p representations of GL2(F) admits no nonzero invariant linear form whenever at least one factor is infinite-dimensional and irreducible with a central character.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026multilinear,
  title = {{On multilinear forms for mod $p$ representations of $\operatornameGL_2$ over a $p$-adic field}},
  author = {Hao Liu and Haoran Wang and Yuting Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26541},
  eprint = {2610.26541},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26541},
}
RIS
TY  - UNPB
TI  - On multilinear forms for mod p representations of GL_2 over a p-adic field
AU  - Hao Liu
AU  - Haoran Wang
AU  - Yuting Yang
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.26541
UR  - https://arxiv.org/abs/2610.26541
AB  - Let F be a finite extension of ℚ_p, for any prime p, and let L/F be a ramified quadratic extension. We compute the spaces of 𝒪_L^×-equivariant linear forms on the representation ind^G_KZσ/(Φ−λ). The only nonzero cases arise from one-dimensional quotients. Consequently, the restriction of an infinite-dimensional irreducible smooth mod p representation of GL_2(F) with a central character to 𝒪_L^× has no character quotient. We apply this vanishing result to show that a finite tensor product of smooth mod p representations of GL_2(F) admits no nonzero invariant linear form whenever at least one factor is infinite-dimensional and irreducible with a central character.
ER  -