決定橢圓曲線上 L 函數的特殊零點

Abstract

橢圓曲線的狄利克雷級數在 s=1 上的階是大家所感興趣的。Mazur、Tate 和 Teitelbaum 發現一個可以計算經扭變後的橢圓曲線的狄利克雷級數在特殊點 s=1 上 是否為零點的方法。這涉及到計算模符號。本篇論文主要內容是在於整理他們所給 出的方法後，同時提出一個完整的演算法列表計算出扭變後的橢圓曲線的狄利克 雷級數在特殊點 s=1 上是否為零點。

The order of the Dirichlet series of the elliptic curve at s=1 is of interest to everyone. Mazur, Tate, and Teitelbaum found a way to calculate whether the Dirichlet series of the twisted elliptic curve has a zero at the particular point s=1. This involves calculating the modulo symbol. The aim of this paper is to sort out the methods they have given, and at the same time propose a complete list of algorithms to calculate whether the Dirichlet series of the twisted elliptic curve has a zero at the special point s=1.