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

TL;DR

This paper explores invariants of stationary AF-algebras derived from incidence matrices and conjectures a link between these invariants and the torsion subgroup of elliptic curves with complex multiplication.

Contribution

It introduces a new invariant for AF-algebras based on polynomial conditions and conjectures a connection to elliptic curve torsion subgroups.

Findings

Invariant Z^n/p(A) Z^n is preserved under strong stable isomorphism.

Conditions on polynomial p(x) determine when the invariant applies.

Conjectured relationship between invariants and elliptic curve torsion subgroups.

Abstract

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.