Invariants of stationary AF-algebras and torsion subgroup of elliptic curves with complex multiplication

Igor Nikolaev

Published 2008-11-26, updated 2013-10-05Version 6

Let G(A) be an AF-algebra given by periodic Bratteli diagram with the incidence matrix A in GL(n, Z). For a given polynomial p(x) in Z[x] we assign to G(A) a finite abelian group Z^n/p(A) Z^n. It is shown that if p(0)=1 or p(0)=-1 and Z[x]/(p(x)) is a principal ideal domain, then Z^n/p(A) Z^n 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.