Coleman積分及p-adic L-函數

Abstract

本文探討在一維p-adic射影空間上的積分理論，並將其使用於建構 p-adic完備域上的對數F-crystal，主要關注在過程中自然構造出的對數多項式函數。對數多項式函數可實際應用在計算p-adic上的L-函數特殊值；準確來說，對數多項式函數在分圓點上的取值和久保田-Leopold L-函數在正整數上的特殊值有連繫。文章以推導Coleman(從Koblitz證明k=1的情形為推廣對象)描述當k為正整數時，L-函數的特殊值以及k次對數多項式關係的公式總結。

In this article we discuss the integration theory on p-adic projective space of dimension 1, and apply it to construct the logarithmic F-crystal on the p-adic complete field, where polylogarithm functions occurs in a natural development. The usage of polylogarithms realize in the computation for the p-adic L-values. To be precise, valuation of the polylogarithms at primitive roots of unity is related to the special values of the Kubota-Leopold L-function at positive integers. Eventually, we conclude by deriving a formula relating the evaluation of p-adic L-functions at k to the k-th polylogarithm, which extends the formula by Koblitz, who proved the case k=1.