Missouri Journal of Mathematical Sciences

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

Igor Nikolaev

Full-text: Open access

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.

Article information

Source
Missouri J. Math. Sci., Volume 26, Issue 1 (2014), 23-32.

Dates
First available in Project Euclid: 10 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1404997106

Digital Object Identifier
doi:10.35834/mjms/1404997106

Mathematical Reviews number (MathSciNet)
MR3229946

Zentralblatt MATH identifier
1364.11113

Subjects
Primary: 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22]
Secondary: 46L85: Noncommutative topology [See also 58B32, 58B34, 58J22]

Keywords
AF-algebras elliptic curves

Citation

Nikolaev, Igor. Invariants of Stationary AF-Algebras and Torsion Subgroups of Elliptic Curves with Complex Multiplication. Missouri J. Math. Sci. 26 (2014), no. 1, 23--32. doi:10.35834/mjms/1404997106. https://projecteuclid.org/euclid.mjms/1404997106


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