Galois action on motivic Zeta functions

• Applicant: Dr. Annabelle Hartmann
• Subject Area: Mathematics
• Term: from 2013 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 244578707

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.

• DFG Programme: Research Fellowships
• International Connection: Belgium
• Participating Institution: Katholieke Universiteit Leuven ; Department of Mathematics
• Host: Professor Dr. Johannes Nicaise