p-adic Rigidity and Supercongruences at Rank Two Attractors

Abstract

We study a p-adic rigidity phenomenon suggested by rank two attractors. Near an ordinary special fiber of a Calabi--Yau family, the excellent Frobenius of Beukers and Vlasenko acts on the parameter disk. Achinger--Zdanowicz construct a canonical W₂-lift of the Frobenius twist, while Brantner--Taelman construct a canonical formal lift. We conjecture that the Dwork and Achinger--Zdanowicz first-order obstruction maps agree and that the fixed points of the excellent Frobenius are precisely the parameters of the Brantner--Taelman canonical formal lifts. Our main theorem gives a cohomological criterion in arbitrary dimension. Under its hypotheses, a Frobenius-stable rank two factor of the p-adic cohomology containing the holomorphic class forces that class to span a Cartier-stable line and gives supercongruences modulo p^2s for every s≥1. Moreover, if the hypotheses of the Beukers--Vlasenko excellent lift theorem hold, then the excellent Frobenius fixes the corresponding parameter. We apply the criterion to the Hulek--Verrill A₄ family at t_*=−1/7, a parameter predicted to be a rank two attractor. Assuming Dummigan's Conjecture~1.1, we prove that the excellent Frobenius fixes t_*, the supercongruences hold, and the fiber with parameter t_*bmod p² is the Achinger--Zdanowicz canonical lift at every prime where the required Dwork ordinarity conditions are satisfied. A CM fiber of the mirror quartic K3 pencil gives a second application. We also prove that the q-power map on a split formal torus has the unit section as its unique periodic point in the identity residue disk over every finite extension of W(𝔽_q)[1/p], where q is the cardinality of the residue field.

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BibTeX RIS

BibTeX

@misc{fu2026padic,
  title = {{p-adic Rigidity and Supercongruences at Rank Two Attractors}},
  author = {Yu Fu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.36045},
  eprint = {2609.36045},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36045},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - p-adic Rigidity and Supercongruences at Rank Two Attractors
AU  - Yu Fu
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.36045
UR  - https://arxiv.org/abs/2609.36045
AB  - We study a p-adic rigidity phenomenon suggested by rank two attractors. Near an ordinary special fiber of a Calabi--Yau family, the excellent Frobenius of Beukers and Vlasenko acts on the parameter disk. Achinger--Zdanowicz construct a canonical W_2-lift of the Frobenius twist, while Brantner--Taelman construct a canonical formal lift. We conjecture that the Dwork and Achinger--Zdanowicz first-order obstruction maps agree and that the fixed points of the excellent Frobenius are precisely the parameters of the Brantner--Taelman canonical formal lifts. Our main theorem gives a cohomological criterion in arbitrary dimension. Under its hypotheses, a Frobenius-stable rank two factor of the p-adic cohomology containing the holomorphic class forces that class to span a Cartier-stable line and gives supercongruences modulo p^2s for every s≥1. Moreover, if the hypotheses of the Beukers--Vlasenko excellent lift theorem hold, then the excellent Frobenius fixes the corresponding parameter. We apply the criterion to the Hulek--Verrill A_4 family at t_*=−1/7, a parameter predicted to be a rank two attractor. Assuming Dummigan's Conjecture~1.1, we prove that the excellent Frobenius fixes t_*, the supercongruences hold, and the fiber with parameter t_*bmod p^2 is the Achinger--Zdanowicz canonical lift at every prime where the required Dwork ordinarity conditions are satisfied. A CM fiber of the mirror quartic K3 pencil gives a second application. We also prove that the q-power map on a split formal torus has the unit section as its unique periodic point in the identity residue disk over every finite extension of W(𝔽_q)[1/p], where q is the cardinality of the residue field.
ER  -

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