We study the interplay between noncommutative tori and noncommutative
elliptic curves through a category of equivariant differential modules on
C∗. We functorially relate this category to the category of
holomorphic vector bundles on noncommutative tori as introduced by Polishchuk
and Schwarz and study the induced map between the corresponding K-theories. In
addition, there is a forgetful functor to the category of noncommutative
elliptic curves of Soibelman and Vologodsky, as well as a forgetful functor to
the category of vector bundles on C∗ with regular singular
connections.
The category that we consider has the nice property of being a Tannakian
category, hence it is equivalent to the category of representations of an
affine group scheme. Via an equivariant version of the Riemann-Hilbert
correspondence we determine this group scheme to be (the algebraic hull of)
Z2. We also obtain a full subcategory of the category of
holomorphic bundles of the noncommutative torus, which is equivalent to the
category of representations of Z. This group is the proposed
topological fundamental group of the noncommutative torus (understood as a
degenerate elliptic curve) and we study Nori's notion of \'etale fundamental
group in this context.Comment: 22 pages with major revisions. Some preliminary material removed.
Section 4 on the \'etale fundamental group of noncommutative tori is entirely
new. References changed accordingly, to appear in JNC