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Meetings of the St. Petersburg Mathematical Society
September 9, 2003, St. Petersburg

Noncommutative geometry, nonstandard mathematics, and the theory of elliptic curves with “real multiplication”

I. B. Fesenko


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Abstract: In the last years methods of operator and von Neumann algebras and more generally so called noncommutative geometry have been applied in a number of works to the study of (rather algebraic structures) of commutative number theoretical objects: e.g. left uncompleted work of Connes on Riemann's zeta function, the works of Manin and Marcolli on Arakelov geometry and modular symbols, the work of Manin on hypothetical theory of real multiplication (Manin's Alterstraum). The last work is a research project aimed to use quantum torii (or quantum degenerate elliptic curves) and quantum theta functions to construct analogues of parts of the classical theory of elliptic curves with complex multiplication.
The talk will first present some of the main structrues of and ideas in those works. Then a new approach to study the number theoretical objects, which unlike the previous, works at the level of arithmetic structures too, will be described. The approach is based on using hyper objects (e.g. hyper elliptic curves with hyper complex multiplication) and the shadow (standard path) map to descend to ordinary objects. It revives some of old ideas of Robinson and Weil. As an applications of hyperdiscretization principle, “noncommutative” spaces can be studied via covering them by hyper commutative spaces. A relation between the former and the shadow image of the latter is then given by a mapping, which could be viewed as a vast generalization of the Seiberg-Witten map in string theory.

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