A criterion for elliptic curves with second lowest 2-power in L(1)
Published online by Cambridge University Press: 26 November 2001
Abstract
Let D = p1 … pm, 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 = x3 − D2x 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
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 131 , Issue 3 , November 2001 , pp. 385 - 404
- Copyright
- © 2001 Cambridge Philosophical Society
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