Nikolaev's torsion conjecture for CM elliptic curves

Let ECME(K)E_{CM}\cong E(K) be an elliptic curve with complex multiplication. Let Etors(K)E_{tors}(K) be the finite-order subgroup of E(K)E(K). If KKK'\supset K is a non-trivial extension obtained by adjoining roots of a polynomial p(x)Z[x]p(x)\in {\mathbb Z}[x] with p(0)=±1p(0)=\pm 1, and GAG_A is the AFAF-algebra associated to ECME_{CM} by the Teichmüller functor, define

Abx1(GA)=Zn/(AI)Zn,Ab_{x-1}(G_A)={\mathbb Z}^n/(A-I){\mathbb Z}^n,

and Abp(x)(GA)=Zn/p(A)ZnAb_{p(x)}(G_A)={\mathbb Z}^n/p(A){\mathbb Z}^n. Nikolaev's torsion conjecture.

Etors(K)Abx1(GA),Etors(K)Abp(x)(GA).E_{tors}(K)\cong Ab_{x-1}(G_A),\qquad E_{tors}(K')\cong Ab_{p(x)}(G_A).

This proposes that torsion points over the base field and over the specified extensions are captured by the corresponding invariants of the associated stationary AFAF-algebra. The source gives no resolution status for this conjecture.

Sources & referencesView supporting material

Primary source

Igor Nikolaev, “Invariants of stationary AF-algebras and torsion subgroup of elliptic curves with complex multiplication”, arXiv:0811.4336 (2013).

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