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A criterion for elliptic curves with lowest 2-power in L(1)

Published online by Cambridge University Press:  01 May 1997

CHUNLAI ZHAO
CHUNLAI ZHAO
Affiliation:
Department of Mathematics, Peking University, Beijing 100871, People's Republic of China
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Abstract

Let D=π1..., πn, where π1, ... πn are distinct Gaussian primes ≡ (mod 4) and n is any positive integer. In this paper, we prove that the value of the Hecke L-function attached to the elliptic curve ED2: y2=x3−D2x at s=1, divided by the period ω defined below, is always divisible by 2n−1. Moreover, we give a simple combinatorial criterion for this value to be exactly divisible by 2n−1. Our results are in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer, and, when D is rational, enable us to prove the conjecture of Birch and Swinnerton-Dyer for ED2 when the value at s=1 is exactly divisible by 2n−1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 121 , Issue 3 , May 1997 , pp. 385 - 400
Copyright
Cambridge Philosophical Society 1997

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Footnotes

This work was supported by NSFC.