椭圆曲线L(1)的 3-adic 赋值达到下界的判别

设 K 为三次分圆域,E _D^lambda]] :y²=x³-2⁴ 3 ³ D^lambda]为定义在 K 上的椭圆曲线, 其中 D 为 K 中无平方因子的代数整数, lambda]属于Z . 邱德荣用一个非常复杂的函数给出了当 lambda]=2,4 时E _D^lambda] 的 Hecke L- 函数在点 1 处的值(除以椭圆周期) 恰被 3 ^{n/2 -1} 整除 的充要条件. 本文用加权图的语言给出了这个充要条件一个简单的组合论的刻画.

Criterion for Elliptic Curves with the Lowest 3-Power in L(1)(Chinese)

Let K be the 3^rd cyclotomic field, E _D^lambda] :y²=x³-2⁴ 3 ³ D^lambda]] be an elliptic curve defined over K, where D\in K is a square-free algebraic integer and lambda]] in Z . Qiu Derong found a rather complicated function as the criterion for the value of the Hecke L-function attached to the curve E _D^lambda](lambda]=2,4) at s=1 divided by the period OMEGA] to be exactly divisible by 3 ^{n/2 -1} . This paper reduced Qiu’s criterion to a simpler version in terms of weighted graphs.