Clifford Multiplication on Spinor Abelian Varieties
Ivona Grzegorczyk, Ricardo Suarez
TL;DR
The paper builds a bridge between complex Clifford algebras and polarized Abelian varieties by constructing spinor Abelian varieties $S_{\Delta}$ from unitary spinor modules $\Delta$ for ${\mathbb C}_{q}(V)$. It proves that ${\mathbb C}_{q}(V)_{\mathbb{Z}} \cong \operatorname{End}(S_{\Delta})$ and that the dual variety $\operatorname{Pic}^{0}(S_{\Delta})$ inherits a spinor structure with a compatible Clifford action, yielding a PPAV structure on the dual. A central result is the full decomposition of $S_{\Delta}$ as $E_{i}^{\times 2^{k}}$, where $E_{i}$ is the CM elliptic curve with $j=1728$, and the automorphism group corresponds to the Clifford generator group. These findings establish explicit endomorphism and decomposition patterns, enabling a spinor-AV interpretation of Clifford actions and suggesting avenues for further exploration in singular Jacobians and real/quaternionic-type spinor varieties.
Abstract
We define a spinor Abelian variety $S_Δ$ to be a complex Abelian variety whose tangent space at the origin is a space of spinors for a suitable complex Clifford algebra $\mathbb{C}_{q}(V)$. We examine intrinsic properties of such varieties and the connection between Clifford multiplication and their endomorphism algebras. We then extend the analysis of Clifford multiplication to the dual torus $Pic^{0}(S_Δ)$.