函数域的Bost-Connes型系统
Bost-Connes type systems for function fields
摘要：
我们描述了一种结构，它关联到任何功能字段$k$和$k$的任何地方$\infty$的$C^*$ -动力系统$(C_{k,\infty},σ_t)$，这是类似于Bost-Connes系统相关联的$\QQ$和它的阿基米德的地方。我们的建设依赖于Hayes的显式类域理论的符号规范化秩一Drinfel的模块。我们证明了$C_{k,\infty}$具有$\Gal(K/k)$的忠实连续作用，其中$K$是Hayes构造的某个域，使得$k^\abi\subset K\subset k^\ab$，其中$k^\abi$是$k$在$\infty$处完全分裂的最大阿贝尔扩张.我们对$(C_{k,\infty},σ_t)$在任何温度下的极端KMS $_β$态进行了分类，并表明在温度为$0<1/β<\infty$时发生了具有自发对称性破缺的相变。在高温$1/β\geqslant 1$，存在唯一的KMS $_β$状态。在低温$1/β<1$下，KMS $_β$极端态的空间在$\Gal(K/k)$下是基本均匀的.每个这样的状态是$\I_\infty$类型，并且配分函数是戴德金ζ函数$ζ_{k,\infty}$。此外，我们构造了一个“有理”*-子代数$\HH$，给出了$\HH$和$C_{k,\infty}$的表示，并证明了低温极端KMS $_β$态在$\HH$的某些元素处的值与偏zeta函数的特殊值有关. \n勘误表：本文错误地认为，在高温$1/β\geqslant 1$下，KMS的唯一$_β$态是$\III_{q^{-β}}$型，其中$q$是$k$的常数子域的基数。Neshveyev和Rustad \cite{NesRus12}证明了正确的类型是$\III_{q^{-βd_\infty}}$，其中$d_\infty$是位置$\infty$的度数。原始声明已被保留以供参考，但在它们旁边插入了勘误表。
英文摘要：
We describe a construction which associates to any function field $k$ and any place $\infty$ of $k$ a $C^*$-dynamical system $(C_{k,\infty},σ_t)$ that is analogous to the Bost-Connes system associated to $\QQ$ and its archimedian place. Our construction relies on Hayes' explicit class field theory in terms of sign-normalized rank one Drinfel'd modules. 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 $\infty$. We classify the extremal KMS$_β$ states of $(C_{k,\infty},σ_t)$ at any temperature $0<1/β<\infty$ and show that a phase transition with spontaneous symmetry breaking occurs at temperature $1/β=1$. At high temperature $1/β\geqslant 1$, there is a unique KMS$_β$ state. At low temperature $1/β<1$, the space of extremal KMS$_β$ states is principal homogeneous under $\Gal(K/k)$. Each such state is of type $\I_\infty$ and the partition function is the Dedekind zeta function $ζ_{k,\infty}$. 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$_β$ 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 $1/β\geqslant 1$, the unique KMS$_β$ state is of type $\III_{q^{-β}}$, where $q$ is the cardinal of the constant subfield of $k$. It has been shown by Neshveyev and Rustad \cite{NesRus12} that the correct type is $\III_{q^{-βd_\infty}}$ where $d_\infty$ is the degree of the place $\infty$. The original statements have been kept for reference, but errata have been inserted next to them.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
