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

Published online by Cambridge University Press:  26 November 2001

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

Abstract

Let D = p1pm, where p1, …, pm are distinct rational primes ≡ 1(mod 8), and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the Hecke L-function of the congruent elliptic curve ED2 : y2 = x3D2x at s = 1, divided by the period ω defined below, to be exactly divisible by 4m. As a corollary, we obtain a series of non-congruent numbers whose number of prime factors tends to infinity, and for which the corresponding elliptic curves have non-trivial 2-part of Tate–Shafarevich group, which greatly generalizes a result of Razar [8]. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer.

Type
Research Article
Copyright
© 2001 Cambridge Philosophical Society

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Footnotes

This work was supported by NSFC and RFDP.