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

Abstract

Let $D\,{=}\,\pi_1\cdots \pi_n$, where $\pi_1,\dots,\pi_n$ are distinct Gaussian odd primes and $n$ is any positive integer. Let $E_{4D^2}$ be the elliptic curve $y^2\,{=}\,x^3-(2D)^2x$. We prove that the value at $s\,{=}\,1$ of the complex $L$-function of $E_{4D^2}$, divided by its natural period(which is $\omega/ \surd{(2D)}$, where $\omega\,{=}\,2{\cdot}6220575\dots$), is always divisible by $2^{n-1}$, and we give a simple combinatorial criterion for it to be exactly divisible by $2^{n-1}$. As a corollary, we construct a series of even non-congruent numbers having an arbitrarily large number of prime factors. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton–Dyer.