A Criterion for Elliptic Curves with Second Lowest 2–Power in L(1) (II)

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.