Real Multiplication and noncommutative geometry
By Yuri I. Manin. 2002
123 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/673307f83d041961869c17ab1141568df2780006
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Noncommutative geometry, dynamics, and ∞-adic Arakelov geometry (2004, background): “In [ 26 ] [27] Manin conjectured that, in the case of real quadratic fields, noncommutative geometry should produce Stark numbers and Galois action via the theory of noncommutative tori.”
- Noncommutative two-tori with real multiplication as noncommutative projective varieties (2002, unknown): “A G ] 2 2 D ec”
- Topics in Noncommutative Geometry (2015, background): “Unfortunately, it has non-trivial SMEBs only for special values of θ (it must be a real quadratic irrationality, corresponding to a noncommutative torus with “real multiplication” [47]).”