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The q-deformed cross-ratio: modular invariants and Coxeter friezes

  • Valentin Ovsienko
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Abstract

We introduce and study a scalar q-deformation of the cross-ratio on ℙ1(ℚ). Our construction is based on the notion of q-deformed rational numbers due to Morier-Genoud and the author. The q-cross-ratio is invariant under PSL(2,ℤ), while elements of determinant −1 of PGL(2,ℤ) act by q↦ q−1. A principal result is its relation to q-deformed Coxeter friezes associated with rational polygons. The expansion at q=eh yields an algebraically independent sequence of modular invariants and relative invariants, although this sequence does not separate modular orbits. We compute the first two nonconstant coefficients of this expansion explicitly.

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Cite

BibTeXRIS
BibTeX
@misc{ovsienko2026qdeformed,
  title = {{The $q$-deformed cross-ratio: modular invariants and Coxeter friezes}},
  author = {Valentin Ovsienko},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10999},
  eprint = {2609.10999},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10999},
}
RIS
TY  - UNPB
TI  - The q-deformed cross-ratio: modular invariants and Coxeter friezes
AU  - Valentin Ovsienko
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.10999
UR  - https://arxiv.org/abs/2609.10999
AB  - We introduce and study a scalar q-deformation of the cross-ratio on ℙ^1(ℚ). Our construction is based on the notion of q-deformed rational numbers due to Morier-Genoud and the author. The q-cross-ratio is invariant under PSL(2,ℤ), while elements of determinant −1 of PGL(2,ℤ) act by q↦ q^−1. A principal result is its relation to q-deformed Coxeter friezes associated with rational polygons. The expansion at q=e^h yields an algebraically independent sequence of modular invariants and relative invariants, although this sequence does not separate modular orbits. We compute the first two nonconstant coefficients of this expansion explicitly.
ER  -