某些椭圆曲线的L－级数的L（1）的非零性

The Nonvanishing of L(1) for the L-series of Some Elliptic Curves

本文考虑了椭圆曲线Γ＿Ｄ：Ｘ￣３＋Ｙ￣３＝ＤＺ￣３．以ＬＤ（ｓ）记Γ＿Ｄ的ＨｅｃｋｅＬ－级数．由Ｌ＿Ｄ（ｓ）的解析延拓我们将Ｌ＿Ｄ（１）展成有限项之和，然后通过建立ｙ￣２＝ｘ￣３－１６的一个处处有好的约化的模型，证明了当ｐ≡２或５（ｍｏｄ ９）时，Ｌ＿（ｐ￣２）（１）≠０．这些结果是对Ｂｉｒｃｈ和Ｓｗｉｎｎｅｒｆｏｎ－Ｄｙｅｒ猜想的支持。

In this paper,we consider the elliptic curvesΓ_D:X￣3+Y￣3=DZ￣3. Let L_D(s)be the Hecke L-series of Γ_D. Using the analytic continuation of L_D(s),we expand L_D(1)as a sum of finite terms. Then by establishing y￣2=x￣3-16 as a model which has good reduction everywhere we prove that for p≡2,5(mod 9),L_p(1)≠0 and L_p￣2(1)≠0.These results may give:some supportable evidences of Birch and Swinnerton-Dyer conjecture。