$\F_1$的乐趣
Fun with $\F_1$
摘要：
我们证明了量子统计力学系统Bost-Connes的代数和内驱力是通过将标量从“一元场”推广到有理数而自然产生的。内动子的阿贝尔部分的归纳结构对应于该“域”的有限扩张的塔，而自同态反映了Frobenius对应。这给出了特别是一个明确的模型在整数这个endomotive，这是有关的原始Hecke代数描述。本文研究了内子代数和相应的非交换交叉积代数在素数处的约化。
英文摘要：
We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost--Connes naturally arises by extension of scalars from the "field with one element" to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that "field", while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra.
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。
