8,355 research outputs found
Noncommutative Geometry and Arithmetic
This is an overview of recent results aimed at developing a geometry of
noncommutative tori with real multiplication, with the purpose of providing a
parallel, for real quadratic fields, of the classical theory of elliptic curves
with complex multiplication for imaginary quadratic fields. This talk
concentrates on two main aspects: the relation of Stark numbers to the geometry
of noncommutative tori with real multiplication, and the shadows of modular
forms on the noncommutative boundary of modular curves, that is, the moduli
space of noncommutative tori. To appear in Proc. ICM 2010.Comment: 16 pages, LaTe
Class invariants for quartic CM fields
One can define class invariants for a quartic primitive CM field K as special
values of certain Siegel (or Hilbert) modular functions at CM points
corresponding to K. We provide explicit bounds on the primes appearing in the
denominators of these algebraic numbers. This allows us, in particular, to
construct S-units in certain abelian extensions of K, where S is effectively
determined by K. It also yields class polynomials for primitive quartic CM
fields whose coefficients are S-integers.Comment: 14 page
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