Abstract
Let D = p1p2 · · ·pm, where p1, p2, . . . ,pm are distinct rational primes with p1 ≡ p2 ≡3(mod 8), pi ≡1(mod 8)(3 ≤ i ≤ m), and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the complex L–function of the congruent elliptic curve \( E_{{D^{2} }} :y^{2} = x^{3} - D^{2} x \)at s = 1, divided by the period ω defined below, to be exactly divisible by 22m–2, the second lowest 2–power with respect to the number of the Gaussian prime factors of D. As a corollary, we obtain a new series of non–congruent numbers whose prime factors can be arbitrarily many. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton–Dyer.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.