The Absolute Twistor Line and the Geometry of Spec Z̅
Abstract
We construct the absolute algebraic geometry of the compactification Spec Z̅ by amalgamating the affine absolute curve (Spec Z)F1 with an archimedean component defined over the signed extension F12 of F1. By adjoining a formal imaginary unit to the absolute projective line, we obtain an equivariant topos endowed with a canonical geometric inversion symmetry, which induces the twistor real structure on its complex points. This archimedean geometry is incorporated into a global absolute curve defined as an internal object of the odd arithmetic topos, dual to the multiplicative monoid of odd positive integers, and governed by the intrinsic Hopf structure of spherical F12-algebras. The restriction of the absolute Frobenius action to odd integers is forced arithmetically by the extension of scalars to F12. On complex points, the resulting dynamics simultaneously generates the Adams operations and complex conjugation on real Hodge structures. At the categorical level, the odd arithmetic topos originates in the pericyclic category, whose λ-operations provide a conceptual interpretation of the local factors of geometric L-functions.
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BibTeX
@misc{connes2026absolute,
title = {{The Absolute Twistor Line and the Geometry of $\overline\textSpec\, \mathbf Z$}},
author = {Alain Connes and Caterina Consani},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00299},
eprint = {2609.00299},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00299},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Absolute Twistor Line and the Geometry of Spec Z̅
AU - Alain Connes
AU - Caterina Consani
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00299
UR - https://arxiv.org/abs/2609.00299
AB - We construct the absolute algebraic geometry of the compactification Spec Z̅ by amalgamating the affine absolute curve (Spec Z)_F_1 with an archimedean component defined over the signed extension F_1^2 of F_1. By adjoining a formal imaginary unit to the absolute projective line, we obtain an equivariant topos endowed with a canonical geometric inversion symmetry, which induces the twistor real structure on its complex points. This archimedean geometry is incorporated into a global absolute curve defined as an internal object of the odd arithmetic topos, dual to the multiplicative monoid of odd positive integers, and governed by the intrinsic Hopf structure of spherical F_1^2-algebras. The restriction of the absolute Frobenius action to odd integers is forced arithmetically by the extension of scalars to F_1^2. On complex points, the resulting dynamics simultaneously generates the Adams operations and complex conjugation on real Hodge structures. At the categorical level, the odd arithmetic topos originates in the pericyclic category, whose λ-operations provide a conceptual interpretation of the local factors of geometric L-functions.
ER -