A conjecture on the torsion points of elliptic curves with the complex multiplication

Details A conjecture on the torsion points of elliptic curves with the complex multiplication A conjecture on the torsion points of elliptic curves with the complex multiplication

2008-11-26

arxiv.org/abs/0811.4336v4

Mathematics

Number Theory

15 pages, 1 figure; an update of the reciprocity conjecture

Scientific paper

Let G(A) be an AF-algebra given by a periodic Bratteli diagram with the incidence matrix A. Depending on a polynomial p(x) in Z[x], we assign to G(A) a finite abelian group Ab [p(x)](G(A))=Z^n/p(A) Z^n. It is shown that for every p(x), such that p(0)=1 or p(0)=-1, the Ab [p(x)] (G(A)) is an invariant of the strong stable isomorphism class of the AF-algebra G(A). Using a categorical correspondence between the elliptic curves and the AF-algebras, a conjecture on the torsion points of an elliptic curve with the complex multiplication is formulated.

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