A dynamical systems analogue of Lichtenbaum's conjectures on special values of Hasse-Weil zeta functions
A dynamical systems analogue of Lichtenbaum's conjectures on special values of Hasse-Weil zeta functions
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Lichtenbaum 对 Hasse-Weil zeta 函数特殊值猜想的动力系统模拟
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发表时间：
2006
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通讯作者：
C. Deninger
中科院分区：
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作者：
C. Deninger
近年来，Lichtenbaum用Weil-\'etale上同调性描述了Hasse-Weil zeta函数的特殊值.在早期的论文中，我们研究了一类与算术方案有一些相似之处的叶状动力系统。假设某些zeta正则化行列式存在，我们现在证明了Lichtenbaum猜想对于一类特别简单的此类动力系统的类似物：使用Alvarez L\'opez和Kordyukov的结果，我们表示Ruelle zeta函数$\zeta^*_R（0）$的主导系数$\zeta ^*_R（0）$在$s = 0$时的解析挠率。然后我们应用Cheeger-M“uller定理用关于调和基的Reidemeister挠率代替解析挠率.对于我们的动力系统，后者可以用与Lichtenbaum的结构相同的配方来表示。
In recent years Lichtenbaum has conjectured a description for the special values of Hasse--Weil zeta functions in terms of ``Weil-\'etale cohomology''. In earlier papers we studied a class of foliated dynamical systems which had some similarities with arithmetic schemes. Assuming that certain zeta regularized determinants exist we now prove an analogue of Lichtenbaum's conjectures for a particularly simple class of such dynamical systems: Using results of \'Alvarez L\'opez and Kordyukov we express the leading coefficient $\zeta^*_R (0)$ of the Ruelle zeta function $\zeta_R (s)$ at $s = 0$ in terms of analytic torsion. We then apply the Cheeger--M\"uller theorem to replace the analytic torsion by the Reidemeister torsion with respect to harmonic bases. For our dynamical systems the latter can be expressed by the same recipe as the one in Lichtenbaum's conjectures.