In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerated surface singularity. We start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f in Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerated with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of the complex zero locus of f close to the origin.
Essentially the entire paper is dedicated to proving that, for f as above, certain candidate poles of Igusa's p-adic zeta function of f, arising from so-called B_1-facets of the Newton polyhedron of f, are actually not poles. This turns out to be much harder than in the topological setting. The combinatorial proof is preceded by a study of the integral points in three-dimensional fundamental parallelepipeds. Together with the work of Lemahieu and Van Proeyen, this main result leads to the Monodromy Conjecture for the p-adic and motivic zeta function of a non-degenerated surface singularity.
Monodromy eigenvalues and zeta functions with differential forms
对于复多项式或解析函数f，人们一直在深入研究其所谓的局部zeta函数或复幂;这些是被认为是s中的函数的|f| ^{2s}w的积分，其中w是具有紧支集的微分形式。它们的极点与f的局部单值特征值之间有很强的对应关系。特别是巴莱表明，每个monodromy特征值的f是形式exp（a2 i π），其中a是这样一个极点。我们证明了类似的p-adic复杂的权力，称为Igusa（本地）zeta函数，但主要是为相关的代数几何拓扑和motivic zeta函数的类似结果。
On Igusa's conjectures for weighted homogeneous polynomials with isolated singularities
KU Leuven（荷语鲁汶大学）；Institute of Mathematics, Vietnam Academy of Science and Technology（越南科学技术院数学研究所）；Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo（东京大学宇宙物理数学研究所（Kavli））；Department of Mathematics, Graduate School of Science, the University of Osaka（大阪大学理学研究科数学系）
On a uniform bound for exponential sums modulo $p^m$ for Deligne polynomials
KU Leuven（荷语鲁汶大学）；Institute of Mathematics, Vietnam Academy of Science and Technology（越南科学技术研究院数学研究所）；Normandie Université, Université de Caen Normandie - CNRS, Laboratoire de Mathématiques Nicolas Oresme (LMNO)（诺曼底大学，卡昂诺曼底大学-法国国家科学研究中心，尼古拉·奥雷数学实验室）
