Paper 4 — Fano Geometry, Automorphisms and the Weil Zeta Function of Cayley–Dickson Zero-Divisors
Abstract
Building on the counting formula |ZD(n)| = 336 · C(n-1,3)₂ for all n ≥ 4, this paper develops the structural theory of zero-divisors in the Cayley–Dickson algebras Aₙ from six directions.
(I) Fano complement geometry. The 7 lo-flats are exactly the complements of the 7 lines of PG(2,F₂); a set {a,b,c,d} is a lo-flat if and only if a⊕b⊕c⊕d = 0.
(II–III) Local and global transitivity. The coboundary-extended stabilizer GL (order 96) acts transitively on the 48 ZD pairs over each lo-flat. Aut(A₃) (order 1344) acts transitively on all 336 pairs in ZD(4), with point-stabilizer (Z/2Z)².
(IV) Automorphism groups. Aut(A₄) ≤ Aut_struct(ZD(4)) (lower bound, order 2688); the exact group is Aut_struct(ZD(4)) = S₇ ≀ G_class, order 7!·768⁷. For all n ≥ 5, the orbit sizes of Aut(Aₙ) on ZD(n) are {2, 14, 84, 336}, classified by the projective geometry of S(z) ⊆ PG(2,F₂), with closed-form count formulas O₃₃₆(n) = 8^(n-4), O₂(n) = |ZD(n-3)|/2, verified through n=7. The image of the shadow map ρ on Type-B axes equals GL(3,F₂) (verified for n=5,...,12).
(V) Fano filtration. ZD(n) = F₄ ⊔ F₅ ⊔ ··· ⊔ Fₙ, where Fₖ consists of ZD pairs first appearing at level k. The Fano bijection f: ZD*(n) → G(3,n-1;F₂) is Aut(Aₙ)-equivariant with uniform fibres of size 336; its Schubert stratification encodes the four orbit types.
(VI) ZD zeta function and Weil identity. The generating function Z(x) satisfies Z(x) = 64x⁴Z(1/8x) and the Weil identity Z(x) = 336x⁴ · Z_Weil(PG(3,F₂),x), equivalent to Poincaré duality of G(3,n-1;F₂).
How to Cite
@misc{levratti2026fano4,
author = {Levratti, Giovanni},
title = {Fano Geometry, Automorphisms and the Weil Zeta Function of Cayley–Dickson Zero-Divisors},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.20417859},
url = {https://doi.org/10.5281/zenodo.20417859},
note = {Paper 4 in the series Zero-Divisors
in Cayley--Dickson Algebras, v10}
}