Galois action on motivic Zeta functions

伊古萨单性猜想预测了多项式f的p进Zeta函数的极点与f的米尔诺纤维的奇异上同调上的单性作用的特征值之间的联系。这是一个相当令人惊讶的猜想，因为第一个猜想与f的计数解密切相关，第二个猜想与f的复奇点的拓扑密切相关，即它将算术和复几何联系起来。动机单性猜想推广了伊古萨单性猜想。在这个猜想中，p进Zeta函数被Denef和Loeser的动机Zeta函数所取代。2007年，Loeser和Nicaise证明了可以从局部奇异级数的Mellin变换中恢复动机zeta函数，局部奇异级数是用形式几何和刚性几何中的积分定义的。动机Zeta函数带有伽罗瓦群的群作用，从中可以得到Milnor纤维上同调上的单态作用。对于局部奇异级数，也可以写出这样的群作用。这个项目的目的是检验这种群体行为，特别是证明它是明确定义的，并且它实际上与Denef和Loesers的动机Zeta函数上的行为是一致的。人们期望从这项研究中得到对单性猜想的一些应用。具体计划是推广Halle和Nicaise利用局部奇异级数上的伽罗瓦作用处理阿贝尔变的单猜想的一篇论文。

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.