Hasse–Witt obstructions for the Aₙ root polytopes

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-17
DOI
https://doi.org/10.5281/zenodo.22817325
Primary Topic
Advanced Mathematical Identities
Type
preprint
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preprint

Hasse–Witt obstructions for the Aₙ root polytopes

Eric Yaw
Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Identities
preprint

Hasse–Witt obstructions for the Aₙ root polytopes

Eric Yaw
preprint en

Abstract

The Beukers–Vlasenko theory of Dwork crystals derives Frobenius structures and p-adic zeta values for Calabi–Yau families whose Newton polytopes have simplicial faces of volume one. The Aₙ root polytopes of the equal-mass banana amplitudes violate that hypothesis because their two-dimensional faces include squares, and I prove that the violation is essential. For every prime p ≥ 5 and every n ≥ 3 the second Hasse–Witt determinant of the full polytope has p-adic valuation exactly L(2) + C(n+1, 2)·C(n−1, 2), one power of p above the Beukers–Vlasenko baseline for each square face, so the k-th Hasse–Witt condition fails for 2 ≤ k < p, and the interior crystal fails at level three by the same count. The excess follows a closed form through level four except at p = 5, and depends on the prime thereafter. I locate the failure in the square-center forms, show that they are exact p³-eigenvectors of the local Cartier operator, with eigenvalue C(3p−3, 2p−2)·C(2p−2, p−1)², prove a corrected decomposition theorem that survives the failure modulo a quotient killed by p, and show that the strong decomposition holds when the Cartier image of the crystal equals that of level k modulo pᵏ, verified at depth one at three primes. For A₃ this yields a theorem conditional on the depth-∞ membership of the corrected center forms, proved at depth one at four primes in the companion paper "The three-loop banana crystal: rank one, the window theorem, and depth one" (DOI 10.5281/zenodo.22135653), on an independence hypothesis, and on irreducibility of the Picard–Fuchs operator. What is obstructed is a sufficient criterion, not its conclusion.

Zenodo (CERN European Organization for Nuclear Research)
Advanced Mathematical Identities
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