Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 41 min ago
PreprintarXiv licencearXiv

The period-index conjecture is false for motivic reasons

  • Alexander Perry
Open at sourcePDFDOICite

Abstract

For any integer d ≥ 3 and any algebraically closed field k of characteristic $0$ and transcendence degree at least d−3, or of characteristic p>2 and transcendence degree at least d−2, we construct a d-dimensional variety over k with a Brauer class of period $2$ and index 2d, violating the period-index conjecture; in particular, the period-index conjecture fails in dimension $3$ over Q̅ and Fp(t)̅, and in all dimensions d ≥ 3 over C and Fp((t))̅. To bound the index of Brauer classes from below, we employ an obstruction of motivic nature, which requires proving the nonexistence of integral Hodge or Tate classes satisfying a certain equation modulo $2$. Motivated by these counterexamples, we propose a period-index conjecture with corrections at small primes and, as evidence, prove its Hodge-theoretic counterpart.

Related

  1. The weighted union-set conjecture is false
    Yimeng Wang
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  2. The List Total Colouring Conjecture is False
    Jonathan A. Noel
    arXivPreprintCC0Full textCombinatorics & discrete mathematicsarXiv
  3. Pierce-Birkhoff conjecture is false
    Zehua Lai, Lek-Heng Lim, Junyu Ren
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. The Partial List Colouring Conjecture is False
    Jonathan A. Noel
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Kusner's conjecture is false for p>4
    Nathan Xiong
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{perry2026periodindex,
  title = {{The period-index conjecture is false for motivic reasons}},
  author = {Alexander Perry},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.03684},
  eprint = {2608.03684},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.03684},
}
RIS
TY  - UNPB
TI  - The period-index conjecture is false for motivic reasons
AU  - Alexander Perry
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2608.03684
UR  - https://arxiv.org/abs/2608.03684
AB  - For any integer d ≥ 3 and any algebraically closed field k of characteristic $0$ and transcendence degree at least d−3, or of characteristic p>2 and transcendence degree at least d−2, we construct a d-dimensional variety over k with a Brauer class of period $2$ and index 2^d, violating the period-index conjecture; in particular, the period-index conjecture fails in dimension $3$ over Q̅ and F_p(t)̅, and in all dimensions d ≥ 3 over C and F_p((t))̅. To bound the index of Brauer classes from below, we employ an obstruction of motivic nature, which requires proving the nonexistence of integral Hodge or Tate classes satisfying a certain equation modulo $2$. Motivated by these counterexamples, we propose a period-index conjecture with corrections at small primes and, as evidence, prove its Hodge-theoretic counterpart.
ER  -