论文标题
量子统计力学和模块化曲线的边界
Quantum Statistical Mechanics and the Boundary of Modular Curves
论文作者
论文摘要
限制模块化符号的理论提供了模块化曲线边界的非共同几何模型，该模块曲线的边界除了尖头外，还包括非理性点。作为与不同的持续分数算法相关的非交通空间家族的一部分，构建了与该边界相关的非共同空间，并具有量子统计机械系统的结构。该量子系统家族的两个特殊案例可以解释为与$ gl_2 $的Shimura品种相关的系统的边界和$ SL_2 $的模拟。讨论了该系统家族的KMS状态的结构。在几何情况下，在边界算术元素上评估的基态由尖缘形式的配对和限制模块化符号给出。
The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as part of a family of noncommutative spaces associated to different continued fractions algorithms, endowed with the structure of a quantum statistical mechanical system. Two special cases of this family of quantum systems can be interpreted as a boundary of the system associated to the Shimura variety of $GL_2$ and an analog for $SL_2$. The structure of KMS states for this family of systems is discussed. In the geometric cases, the ground states evaluated on boundary arithmetic elements are given by pairings of cusp forms and limiting modular symbols.
