## Missouri Journal of Mathematical Sciences

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

Igor Nikolaev

#### 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

https://projecteuclid.org/euclid.mjms/1404997106

Digital Object Identifier
doi:10.35834/mjms/1404997106

Mathematical Reviews number (MathSciNet)
MR3229946

Zentralblatt MATH identifier
1364.11113

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

#### References

• O. Bratteli, Inductive limits of finite dimensional $C^*$-algebras, Trans. Amer. Math. Soc., 171 (1972), 195–234.
• O. Bratteli, P. E. T. Jorgensen, and V. Ostrovsky, Representation Theory and Numerical AF-Invariants, Memoirs Amer. Math. Soc., Vol. 168 (2004).
• E. G. Effros, Dimensions and $C^*$-Algebras, Conf. Board of the Math. Sciences, Regional conference series in Math., Vol. 46, AMS, 1981.
• R. Hartshorne, Algebraic Geometry, GTM 52, Springer, New York, 1977.
• Yu. I. Manin, Real multiplication and noncommutative geometry, in “Legacy of Niels Hendrik Abel”, 685–727, Springer, New York, 2004.
• P. J. Morandi, The Smith normal form of a matrix, 2005, available at \tthttp://sierra.nmsu.edu/morandi/notes/SmithNormalForm.pdf
• I. V. Nikolaev, Remark on the rank of elliptic curves, Osaka J. Math., 46 (2009), 515–527.
• L. D. Olson, Points of finite order on elliptic curves with complex multiplication, Manuscripta Math., 14 (1974), 195–205.
• M. A. Rieffel, Non-commutative tori – a case study of non-commutative differentiable manifolds, Contemp. Math., 105 (1990), 191–211. \tthttp://math.berkeley.edu/$\sim$rieffel/
• M. Rørdam, F. Larsen, and N. Laustsen, An introduction to $K$-theory for $C^*$-algebras, London Mathematical Society Student Texts, 49, Cambridge University Press, Cambridge, 2000.
• I. R. Shafarevich, Basic Notions of Algebra, in Algebra I, E.M.S, Vol. 11, Springer, New York, 1990.
• J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, Springer, New York, 1994.
• J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer, New York, 1992.
• J. B. Wagoner, Strong shift equivalence theory and the shift equivalence problem, Bull. Amer. Math. Soc., 36 (1999), 271–296.