Galois action on motivic Zeta functions

Igusa单值猜想预言了多项式f的p-adic Zeta函数的极点与f的Milnor纤维的奇异上同调上的单值作用的特征值之间的联系。这是一个相当令人惊讶的猜想，因为第一个与f的计数解密切相关，第二个与f的复奇点的拓扑结构密切相关，即它连接了算术和复几何。动机单值猜想推广了Igusa单值猜想。在这个猜想中，p-adic Zeta函数被Denef和Loeser的motivic Zeta函数所取代。在2007年，Loeser和Nicaise证明了motivic zeta函数可以从局部奇异级数的Mellin变换中恢复，这是使用形式和刚性几何中的积分定义的。动机Zeta函数伴随着某个伽罗瓦群的群作用，从中可以恢复Milnor纤维上同调的单值作用。这样的群作用也可以写成局部奇异级数。这个项目的目的是检查这个组的行动，特别是要表明，它是很好的定义，它实际上符合行动的动机Zeta功能的Denef和Loesers。人们期望从这项研究中得到一些对单值猜想的应用。具体的计划是推广一份文件的哈勒和Nicaise处理单值猜想的阿贝尔品种的帮助下的伽罗瓦行动的地方奇异系列。

The Igusa monodromy conjecture predicts a connection between the poles of the p-adic Zeta function of a polynomial f and the eigenvalues of the monodromy action on the singular cohomology of the Milnor fiber of f. This is a quite surprising conjecture, because the first is closely connected to counting solutions of f, and the second to the topology of the complex singularities of f, i.e. it connects arithmetic and complex geometry. The motivic monodromy conjecture generalizes the Igusa monodromy conjecture. In this conjecture the p-adic Zeta function is replaced by the motivic zeta function of Denef and Loeser. In 2007, Loeser and Nicaise showed that the motivic zeta function can be recovered from the Mellin transform of the local singular series, which is defined using integration in formal and rigid geometry. The motivic Zeta function comes with a group action of some Galois group, from which one can recover the monodromy action on the cohomology of the Milnor fiber. Such a group action can also be written down for the local singular series. The aim of this project is to examine this group action, in particular to show that it is well defined, and that it actually coincides with the action on the motivic Zeta function of Denef and Loesers. One expects some application towards the monodromy conjecture from this study. The concrete plan is to generalize a paper of Halle and Nicaise dealing with the monodromy conjecture for abelian varieties with the help of the Galois action on the local singular series.