Noncommutative Geometry and Arithmetic
نویسنده
چکیده
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.
منابع مشابه
The Geometry of Arithmetic Noncommutative Projective Lines
Let k be a perfect field and let K/k be a finite extension of fields. An arithmetic noncommutative projective line is a noncommutative space of the form ProjSK(V ), where V be a k-central two-sided vector space over K of rank two and SK(V ) is the noncommutative symmetric algebra generated by V over K defined by M. Van den Bergh [26]. We study the geometry of these spaces. More precisely, we pr...
متن کاملModuli of complex curves and noncommutative geometry: Riemann surfaces and dimension groups
This paper is a brief account of the moduli of complex curves from the perspective of noncommutative geometry. Using a uniformization of Riemann surfaces by the ordered abelian groups, we prove that modulo the Torelli group, the mapping class group of surface of genus g with n holes, is linear arithmetic group of rank 6g − 6 + 2n.
متن کاملNew perspectives in Arakelov geometry
In this paper we give a uni ed description of the archimedean and the totally split degenerate bers of an arithmetic surface, using operator algebras and Connes' theory of spectral triples in noncommutative geometry.
متن کاملNotes on Noncommutative Geometry
Noncommutative geometry has roots in and is a synthesis of a number of diverse areas of mathematics, including: • Hilbert space and single operator theory; • Operator algebras (C*-algebras and von Neumann algebras); • Spin geometry – Dirac operators – index theory; • Algebraic topology – homological algebra. It has certainly also been inspired by quantum mechanics, and, besides feedback to the ...
متن کاملStability of additive functional equation on discrete quantum semigroups
We construct a noncommutative analog of additive functional equations on discrete quantum semigroups and show that this noncommutative functional equation has Hyers-Ulam stability on amenable discrete quantum semigroups. The discrete quantum semigroups that we consider in this paper are in the sense of van Daele, and the amenability is in the sense of Bèdos-Murphy-Tuset. Our main result genera...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010