Mathematical Sciences: Properties of p-adic complex powers

本研究将研究p-adic复幂的性质，特别是p-adic复幂的极点、monodromy的特征值和b-函数的根之间的联系。第一个问题是要表明，在一般情况下，任意尺寸的monodromy的每一个特征值必须与一个极点的p-adic复杂的权力。第二个问题是研究p进复幂的极点与具有拟常奇点的低维曲面的monodromy特征值之间的关系。与此相关的第三个问题是这类曲面的b函数根的几何解释。这项研究是在令人兴奋的领域桥梁数论和代数几何。后一个主题的强大工具被用来解决前一个主题的问题。这项研究是一个新的和博学的领域，有很多的希望。Meuser是这里真实的专家之一，人们只能等着看她揭示了什么重要和意想不到的结果。

This research will study the properties of p-adic complex powers, more specifically the connections between the poles of the p-adic complex power, eigenvalues of monodromy and roots of the b-function. The first problem is to show that in generic cases of arbitrary dimension every eigenvalue of monodromy must be associated to a pole of the p-adic complex power. The second problem is to examine the relationship between poles of the p- adic complex power and eigenvalues of monodromy in the low dimensional case of surfaces with quasi-ordinary singularities. A third problem related to this would be a geometrical interpretation of roots of the b-function for this class of surfaces. This research is in the exciting area which bridges number theory and algebraic geometry. The powerful tools of the latter subject are brought to bear in solving problems in the former. This research is in a new and erudite area with lots of promise. Meuser is one of the real experts here and one can only wait to see what important and unexpected result she uncovers.