Invariants of Stationary AF-Algebras and Torsion Subgroups of Elliptic Curves with Complex Multiplication

Abstract

Let $G_A$ be an $AF$-algebra given by a periodic Bratteli diagram with the incidence matrix $A\in GL(n, {\Bbb Z})$. For a given polynomial $p(x)\in {\Bbb Z}[x]$ we assign to $G_A$ a finite abelian group $Ab_{p(x)}(G_A)={\Bbb Z}^n/p(A){\Bbb Z}^n$. It is shown that if $p(0)=\pm 1$ and ${\Bbb Z}[x]/\langle p(x)\rangle$ is a principal ideal domain, then $Ab_{p(x)}(G_A)$ is an invariant of the strong stable isomorphism class of $G_A$. For $n=2$ and $p(x)=x-1$ we conjecture a formula linking values of the invariant and torsion subgroup of elliptic curves with complex multiplication.