{"paper":{"title":"Bost-Connes type systems for function fields","arxiv_id":"math/0602554","primary_cat":"math.OA","submitted_at":"2006-02-24T19:37:39+00:00","version_compiled":"v5"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v5","sentences":[{"i":0,"p":1.0,"id":"s0","own":true,"role":"contribution","text":"We describe a construction which associates to any function field $k$ and any place $\\infty$ of $k$ a $C^*$-dynamical system $(C_{k,\\infty},\\sigma_t)$ that is analogous to the Bost-Connes system associated to $\\QQ$ and its archimedian place.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.33,"quantitative":false,"quantitative_p":0.4,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":1.0,"interpretation":0.0}},{"i":1,"p":0.99,"id":"s1","own":false,"role":"method","text":"Our construction relies on Hayes' explicit class field theory in terms of sign-normalized rank one Drinfel'd modules.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.35,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.99,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":2,"p":0.83,"id":"s2","own":true,"role":"result","text":"We show that $C_{k,\\infty}$ has a faithful continuous action of $\\Gal(K/k)$, where $K$ is a certain field constructed by Hayes, such that $k^\\abi\\subset K\\subset k^\\ab$, where $k^\\abi$ is the maximal abelian extension of $k$ that is totally split at $\\infty$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.44,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.83,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.17,"interpretation":0.0}},{"i":3,"p":0.89,"id":"s3","own":true,"role":"result","text":"We classify the extremal KMS$_\\beta$ states of $(C_{k,\\infty},\\sigma_t)$ at any temperature $0<1/\\beta<\\infty$ and show that a phase transition with spontaneous symmetry breaking occurs at temperature $1/\\beta=1$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.55,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.89,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.11,"interpretation":0.0}},{"i":4,"p":1.0,"id":"s4","own":true,"role":"result","text":"At high temperature $1/\\beta\\geqslant 1$, there is a unique KMS$_\\beta$ state.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.11,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"result","text":"At low temperature $1/\\beta<1$, the space of extremal KMS$_\\beta$ states is principal homogeneous under $\\Gal(K/k)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.16,"quantitative":true,"quantitative_p":0.77,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":1.0,"id":"s6","own":true,"role":"result","text":"Each such state is of type $\\I_\\infty$ and the partition function is the Dedekind zeta function $\\zeta_{k,\\infty}$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.11,"quantitative":false,"quantitative_p":0.21,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":7,"p":0.4,"id":"s7","own":true,"role":"contribution","text":"Moreover, we construct a \"rational\" *-subalgebra $\\HH$, we give a presentation of $\\HH$ and of $C_{k,\\infty}$, and we show that the values of the low-temperature extremal KMS$_\\beta$ states at certain elements of $\\HH$ are related to special values of partial zeta functions.\\n Erratum: This article wrongly claims that at high temperature $1/\\beta\\geqslant 1$, the unique KMS$_\\beta$ state is of type $\\III_{q^{-\\beta}}$, where $q$ is the cardinal of the constant subfield of $k$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.11,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.23,"background":0.0,"limitation":0.37,"future_work":0.0,"contribution":0.4,"interpretation":0.0}},{"i":8,"p":0.52,"id":"s8","own":false,"role":"limitation","text":"It has been shown by Neshveyev and Rustad \\cite{NesRus12} that the correct type is $\\III_{q^{-\\beta d_\\infty}}$ where $d_\\infty$ is the degree of the place $\\infty$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.48,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.03,"background":0.45,"limitation":0.52,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":9,"p":0.92,"id":"s9","own":true,"role":"limitation","text":"The original statements have been kept for reference, but errata have been inserted next to them.","hedge":{"p":0.97,"label":"asserted"},"own_p":0.56,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.0,"background":0.05,"limitation":0.92,"future_work":0.0,"contribution":0.01,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.8,"text":"We classify the extremal KMS$_\\beta$ states of $(C_{k,\\infty},\\sigma_t)$ at any temperature $0<1/\\beta<\\infty$ and show that a phase transition with spontaneous symmetry breaking occurs at temperature $1/\\beta=1$.","location":"abstract 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