Real Multiplication and noncommutative geometry

Abstract

Classical theory of Complex Multiplication (CM) shows that all abelian extensions of a complex quadratic field $K$ are generated by the values of appropriate modular functions at the points of finite order of elliptic curves whose endomorphism rings are orders in $K$. For real quadratic fields, a similar description is not known. However, the relevant (still unproved) case of Stark conjectures ([St1]) strongly suggests that such a description must exist. In this paper we propose to use two--dimensional quantum tori corresponding to real quadratic irrationalities as a replacement of elliptic curves with complex multiplication. We discuss some basic constructions of the theory of quantum tori from the perspective of this Real Multiplication (RM) research project.

46 pp., amstex file, no figures

References in corpus (4)

• Quantum Dilogarithm
• Morita equivalence and duality
• Finite Quantum Groupoids and Their Applications
• Von Zahlen und Figuren

Cited by in corpus (32)

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• "New" Veneziano amplitudes from "old" Fermat (hyper) surfaces
• Categorical Non-commutative Geometry
• Bost-Connes-Marcolli systems for Shimura varieties. I. Definitions and formal analytic properties
• Noncommutative geometry of algebraic curves
• Notes on noncommutative geometry
• A Category of Spectral Triples and Discrete Groups with Length Function
• On complex and noncommutative tori
• Modules over the Noncommutative Torus and Elliptic Curves
• Three examples of noncommutative boundaries of Shimura varieties
• Modular Invariant of Quantum Tori
• On a symmetry of complex and real multiplication
• Topics in Noncommutative Geometry
• Non-commutative geometry, dynamics, and infinity-adic Arakelov geometry
• Classical gauge theory on quantum principal bundles
• On a Teichmueller functor between the categories of complex tori and the Effros-Shen algebras
• Symmetry of Quantum Torus with Crossed Product Algebra
• Modular Invariant of Quantum Tori II: The Golden Mean
• Riemann surfaces and AF-algebras
• Quantum Thetas on Noncommutative T^4 from Embeddings into Lattice
• Invariants of stationary AF-algebras and torsion subgroup of elliptic curves with complex multiplication
• Elliptic curves and continued fractions
• On a desingularization of the moduli space of noncommutative tori
• Hecke's integral formula for quadratic extensions of a number field
• Measured foliations and Hilbert 12th problem
• Remark on the rank of elliptic curves
• Quantum statistical mechanics and class field Theory
• AF-algebras and topology of 3-manifolds