Rational points on varieties and Morita equivalences of C∗C^*-algebras

Let $V(k)$ be a projective variety over a number field $k\subset\mathbf{C}$ and let $\mathscr{A}_V$ be the Serre $C^*$-algebra of $V(k)$. We construct a functor $F: V(k)\mapsto \mathscr{A}_V$, such that the $\mathbf{C}$-isomorphic ($k$-isomorphic, resp.) varieties $V(k)$ map to the Morita equivalent (isomorphic, resp.) $C^*$-algebras $\mathscr{A}_V$. In other words, the isomorphisms of the algebra $\mathscr{A}_V=F(V(k))$ preserve the $k$-rational points of $V(k)$, while the Morita equivalences of $\mathscr{A}_V$ correspond to the twists of the variety $V(k)$. We apply the result to the arithmetic geometry of the rational elliptic curves.Comment: 7 pages; an updat

Remark on the rank of elliptic curves

A covariant functor on the elliptic curves with complex multiplication is constructed. The functor takes values in the noncommutative tori with real multiplication. A conjecture on the rank of an elliptic curve is formulated.Comment: 13 pages, 2 figures; to appear Osaka J. Mathematics 46 (2009), No.2; http://projecteuclid.org/euclid.ojm/124541568

Riemann surfaces and AF-algebras

For a generic set in the Teichmueller space, we construct a covariant functor with the range in a category of the AF-algebras; the functor maps isomorphic Riemann surfaces to the stably isomorphic AF-algebras. As a special case, one gets a categorical correspondence between complex tori and the so-called Effros-Shen algebras.Comment: to appear Annals of Functional Analysis. arXiv admin note: substantial text overlap with arXiv:math/010407

Noncommutative geometry of algebraic curves

A covariant functor from the category of generic complex algebraic curves to a category of the AF-algebras is constructed. The construction is based on a representation of the Teichmueller space of a curve by the measured foliations due to Douady, Hubbard, Masur and Thurston. The functor maps isomorphic algebraic curves to the stably isomorphic AF-algebras.Comment: 10 pages, final version; to appear Proc. Amer. Math. So