Tree-Lattice Zeta Functions and Class Numbers
Tree-Lattice Zeta Functions and Class Numbers
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DOI：
10.1307/mmj/1529460323
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发表时间：
2014-02
影响因子：
0.9
通讯作者：
A. Deitmar;Ming-Hsuan Kang
中科院分区：
文献类型：
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作者：
A. Deitmar;Ming-Hsuan Kang
将Ihara zeta函数理论推广到Bruhat-Tits树的非紧算术商。这个新的函数是一个有理函数，尽管它是无限维的。通常它有零点和极点，与紧化情况相反。如果将行列式定义为所有有限主次的极限，则Bass和Ihara的行列式公式成立。在此基础上，导出了一个素数测地线定理，并将其应用于特殊算术群，得到了关于全局域阶数的渐近断言。
The theory of Ihara zeta functions is extended to non-compact arithmetic quotients of Bruhat-Tits trees. This new zeta function turns out to be a rational function, despite the infinite-dimensional setting. In general it has zeros and poles, in contrast to the compact case. The determinant formulas of Bass and Ihara hold true if one defines the determinant as limit of all finite principal minors. From this analysis, a prime geodesic theorem is derived, which, applied to special arithmetic groups, yields new asymptotic assertions on class numbers of orders in global fields.