Noncommutative geometry and motives: the thermodynamics of endomotives TeX Export

@misc{citeulike:463992, abstract = {We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of \$L\$-functions. The analogue in characteristic zero of the action of the Frobenius on l-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.}, archivePrefix = {arXiv}, author = {Connes, Alain and Consani, Caterina and Marcolli, Matilde}, citeulike-article-id = {463992}, citeulike-linkout-0 = {http://arxiv.org/abs/math.QA/0512138}, citeulike-linkout-1 = {http://arxiv.org/pdf/math.QA/0512138}, eprint = {math.QA/0512138}, keywords = {differential, galoisgroup, geometry, noncommutative, numbertheory}, month = {Dec}, posted-at = {2006-01-13 06:46:15}, priority = {2}, title = {Noncommutative geometry and motives: the thermodynamics of endomotives}, url = {http://arxiv.org/abs/math.QA/0512138}, year = {2005} }

TY - ELEC ID - citeulike:463992 L3 - citeulike-article-id:463992 N2 - We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of $L$-functions. The analogue in characteristic zero of the action of the Frobenius on l-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties. TI - Noncommutative geometry and motives: the thermodynamics of endomotives KW - differential KW - galoisgroup KW - geometry KW - noncommutative KW - numbertheory AU - Connes, Alain AU - Consani, Caterina AU - Marcolli, Matilde PY - 2005/Dec/07/ UR - http://arxiv.org/abs/math.QA/0512138 ER -