In these lectures we survey some relations between L-functions and the BC-system, including new results obtained in collaboration with C. Consani. For each prime p and embedding σ of the multiplicative group of an algebraic closure of \({\mathbb {F}_p}\) as complex roots of unity, we construct a p-adic indecomposable representation π_σ of the integral BC-system. This construction is done using the identification of the big Witt ring of \({\bar{\mathbb F}_p}\) and by implementing the Artin–Hasse exponentials. The obtained representations are the p-adic analogues of the complex, extremal KMS_∞ states of the BC-system. We use the theory of p-adic L-functions to determine the partition function. Together with the analogue of the Witt construction in characteristic one, these results provide further evidence towards the construction of an analogue, for the global field of rational numbers, of the curve which provides the geometric support for the arithmetic of function fields.

中文翻译：

---

BC系统和L函数

在这些讲座中，我们调查了L函数与BC系统之间的一些关系，包括与C. Consani合作获得的新结果。对于每一个素数p和的代数闭合的乘法群的嵌入σ \（{\ mathbb {F} _p} \）作为统一的复根，我们构建了一个p进制不可分解表示π _σ积分BC-系统的。这种构造是通过识别\（{\ bar {\ mathbb F} _p} \）的大Witt环并实现Artin-Hasse指数来完成的。将所得到的表示是p复杂，极值KMS的进制类似物_∞BC系统的状态。我们使用p -adic L-函数的理论来确定分区函数。这些结果与特征之一中的维特构造的类似物一起，为构造有理数的全局范围的曲线的类似物提供了进一步的证据，该曲线为函数域的算术提供了几何支持。