Definite integral Albert algebras: arithmetic and finite geometry
Abstract
We give a unified treatment of definite Albert algebras over ℤ, with exceptional Siegel--Weil identities as the main arithmetic tool. Starting from the standard octonion order alone, its first scalar theta coefficient saturates the genus average and forces every remaining class to be rootless. A cubic-ring coefficient of the G2 theta lift then gives the mass of the remaining scale-$2$ framed orders, also recovered from shell geometry. A uniform Niemeier--Golay stabilizer bound saturates this mass, proving that there is exactly one further class and determining both automorphism orders, frame transitivity, and the $2457$ frames on $819$ rank-one elements. The associated generalized hexagon identifies the nonstandard group as 3D4(2):3 only after its order has been proved. An independent weighted-theta argument gives the trace-zero root-system alternative ∅ or A2E83. The construction relates positive masses of framed objects to exceptional Eisenstein coefficients.
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BibTeX
@misc{hohn2026definite,
title = {{Definite integral Albert algebras: arithmetic and finite geometry}},
author = {Gerald Höhn},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27550},
eprint = {2609.27550},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27550},
}
RIS
TY - UNPB
TI - Definite integral Albert algebras: arithmetic and finite geometry
AU - Gerald Höhn
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27550
UR - https://arxiv.org/abs/2609.27550
AB - We give a unified treatment of definite Albert algebras over ℤ, with exceptional Siegel--Weil identities as the main arithmetic tool. Starting from the standard octonion order alone, its first scalar theta coefficient saturates the genus average and forces every remaining class to be rootless. A cubic-ring coefficient of the G_2 theta lift then gives the mass of the remaining scale-$2$ framed orders, also recovered from shell geometry. A uniform Niemeier--Golay stabilizer bound saturates this mass, proving that there is exactly one further class and determining both automorphism orders, frame transitivity, and the $2457$ frames on $819$ rank-one elements. The associated generalized hexagon identifies the nonstandard group as ^3D_4(2):3 only after its order has been proved. An independent weighted-theta argument gives the trace-zero root-system alternative ∅ or A_2E_8^3. The construction relates positive masses of framed objects to exceptional Eisenstein coefficients.
ER -