Abstract
We intend to illustrate how the methods of noncommutative geometry are currently used to tackle problems in class field theory. Noncommutative geometry enables one to think geometrically in situations in which the classical notion of space formed of points is no longer adequate, and thus a “noncommutative space” is needed; a full account of this approach is given in [3] by its main contributor, Alain Connes. The class field theory, i.e., number theory within the realm of Galois theory, is undoubtedly one of the main achievements in arithmetics, leading to an important algebraic machinery; for a modern overview, see [23]. The relationship between noncommutative geometry and number theory is one of the many themes treated in [22, 7–9, 11], a small part of which we will try to put in a more down-to-earth perspective, illustrating through an example what should be called an “application of physics to mathematics,” and our only purpose is to introduce nonspecialists to this beautiful area.
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References
J. Bernstein and S. Gelbart, Eds., An Introduction to the Langlands Program (Birkhäuser, 2003).
J.B. Bost and A. Connes, “Hecke Algebras, Type III Factors and Phase Transitions with Spontaneous Symmetry Breaking in Number Theory,” Selecta Math. (N.S.) 1(3), 411–457 (1995).
A. Connes, Noncommutative Geometry (Academic Press, Inc., San Diego, CA, 1994).
A. Connes, “Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function,” Selecta Math. (N.S.) 5(1), 29–106 (1999).
A. Connes, “Noncommutative Geometry, Year 2000,” IHÉS, Preprint (2000).
A. Connes, “A Short Survey of Noncommutative Geometry,” IHÉS, Preprint (2000).
A. Connes and M. Marcolli, “From Physics to Number Theory via Noncommutative Geometry,” in Frontiers in Number Theory, Physics and Geometry. I, Ed. by P. Cartier et al. (Springer, 2006), pp. 249–347; Preprint, arXiv:hep-th/0404128.
A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, AMS Colloquium Publ. 55 (AMS, Providence; Hindustan Book Agency, New Delhi, 2008); ftp://ftp.alainconnes.org/bookjuly.pdf.
A. Connes and M. Marcolli, “Renormalization, the Riemann-Hilbert Correspondence, and Motivic Galois Theory,” in Frontiers in Number Theory, Physics and Geometry II, Ed. by P. Cartier et al. (Springer, 2007); Preprint, arXiv:hep-th/0411114.
A. Connes, M. Marcolli, and N. Ramachandran, “KMS States and Complex Multiplication,” Selecta Math. (N.S.) 11(3–4), 325–347 (2005); Preprint (2005), arXiv:math.OA/0501424.
C. Consani and M. Marcolli, Eds., Noncommutative Geometry and Number Theory, Aspects Math. E 37 (Vieweg, 2006).
S. Gelbart, ”An Elementary Introduction to the Langlands Program,” Bull. Amer. Math. Soc. (N.S.) 10(2), 177–219 (1984).
E. Ha and F. Paugam, “Bost-Connes-Marcolli Systems for Shimura Varieties. Part I,” Int. Math. Res. Papers 5 (2005).
S. Haran, The Mysteries of the Real Prime, London Math. Soc. Monographs (New Series) 25 (The Clarendon Press, Oxford University Press, New York, 2001).
G. J. Janusz, “Algebraic Number Fields,” Graduate Studies in Mathematics 7 (American Mathematical Society, Providence, RI, 1996 (ed. or. 1973)).
J. M. Gracia-Bondía, J.C. Várilly, and H. Figueroa, Elements of Noncommutative Geometry (Birkhäuser, Boston, 2001).
M. Khalkhali, “Very Basic Noncommutative Geometry,” Preprint (2004), arXiv:math.KT/0408416.
H. Koch, “Algebraic Number Fields,” in Encyclopaedia of Mathematical Sciences 62, Number Theory II, Ed. by A. N. Parshin and I. R. Shafarevich (Springer, 1992).
G. Landi, An Introduction to Noncommutative Spaces and Their Geometries (Springer, Berlin, 1998).
G. W. Mackey, Unitary Group Representations, Advanced Books Classics (Addison-Wesley, 1989 (or. ed. 1978)).
Y. I. Manin, “Von Zahlen und Figuren,” Preprint (2002), arXiv:math.AG/0201005.
M. Marcolli, Arithmetic Noncommutative Geometry, Univ. Lecture Ser. 36 (American Mathematical Society, 2005); Preprint, arXiv:math.QA/0409520.
K. Myiake, Ed., Class Field Theory — Its Centenary and Prospect, Advanced Studies in Pure Mathematics, Msj International Research Institute (Tokyo, 1998).
J. Neukirch, Algebraic Number Theory (Springer, 1999).
N. Schappacher, “On the History of Hilbert’s Twelfth Problem, a Comedy of Errors,” in Matériaux pour l’histoire des mathématiques au XXème siècle — Actes du colloque à la mémoire de Jean Dieudonné (Nice 1996), Sémin. Congr. 3 (Société Mathématique de France, 1998).
P. Stevenhagen, “Hilbert’s 12th Problem, Complex Multiplication and Shimura Reciprocity,” in Class Field Theory — Its Centenary and Prospect, Ed. by K. Myiake, Adv. Stud. Pure Math. 30 (Msj International Research Institute, Tokyo, 1998).
J. C. Várilly, An Introduction to Noncommutative Geometry (European Mathematical Society, 2006).
S. Vladut, Kronecker’s Jugendtraum and Modular Functions (Gordon and Breach, 1991).
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To my prime friend Nicolae Teleman on the occasion of his 65th birthday
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Almeida, P. Noncommutative geometry and arithmetics. Russ. J. Math. Phys. 16, 350–362 (2009). https://doi.org/10.1134/S1061920809030030
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DOI: https://doi.org/10.1134/S1061920809030030