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Published July 13, 2017 | Submitted
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From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices

Abstract

Several recent results reveal a surprising connection between modular forms and noncommutative geometry. The first occurrence came from the classification of noncommutative three spheres, [C–DuboisViolette-I] [C–DuboisViolette-II]. Hard computations with the noncommutative analog of the Jacobian involving the ninth power of the Dedekind eta function were necessary in order to analyze the relation between such spheres and noncommutative nilmanifolds. Another occurrence can be seen in the computation of the explicit cyclic cohomology Chern character of a spectral triple on SU_q(2) [C–02]. Another surprise came recently from a remarkable action of the Hopf algebra of transverse geometry of foliations of codimension one on the space of lattices modulo Hecke correspondences, described in the framework of noncommutative geometry, using a modular Hecke algebra obtained as the cross product of modular forms by the action of Hecke correspondences [C–Moscovici-I] [C–Moscovici-II]. This action determines a differentiable structure on this noncommutative space, related to the Rankin–Cohen brackets of modular forms, and shows their compatibility with Hecke operators. Another instance where properties of modular forms can be recast in the context of noncommutative geometry can be found in the theory of modular symbols and Mellin transforms of cusp forms of weight two, which can be recovered from the geometry of the moduli space of Morita equivalence classes of noncommutative tori viewed as boundary of the modular curve [Manin–M].

Additional Information

(Submitted on 6 Apr 2004) We are very grateful to Niranjan Ramachandran for many extremely useful conversations on class field theory and KMS states, that motivated the GL_2 system described here, whose relation to the theory of complex multiplication is being investigated in [13]. We thank Marcelo Laca for giving us an extensive update on the further developments on [5]. We benefited from visits of the first author to MPI and of the second author to IHES and we thank both institutions for their hospitality. The second author is partially supported by a Sofja Kovalevskaya Award of the Humboldt Foundation and the German Government.

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