A criterion for elliptic curves with second lowest 2-power in \(L(1)\) (Q2758153)

Let \(D \in \mathbb Z[i]\) be a squarefree nonunit in the ring of Gaussian integers. Then the elliptic curve \(E_D: y^2 = x^3 - D^2x\) has complex multiplication by \(\mathbb Z[i]\); thus we can attach a grössencharacter \(\psi_D\) to \(E_D\), and we let \(L(\overline{\psi}_D,s)\) denote the Hecke \(L\)-function associated to the conjugate of \(\psi_D\). Finally, let \(\omega\) denote the real period of \(y^2 = 4x^3 - 4x\). Then it is known that \(L(\overline{\psi}_D,1)/\omega\) is a rational number.NEWLINENEWLINEIn [Math. Proc. Camb. Philos. Soc. 121, No. 3, 385--400 (1997; Zbl 0882.11039)], the author proved that if \(D\) is a product of \(n\) distinct Gaussian primes \(\pi_k \equiv 1 \bmod 4\), then \(L(\overline{\psi}_D,1)/\omega\) is divisible by \(2^{n-1}\). In this article, the author gives a graph theoretical criterion for when \(2^{n-1}\) divides this quotient exactly. He also gives applications to curves \(E_D\) where \(D\) is the product of rational primes \(\equiv 1 \bmod 8\), adding support to the conjecture of Birch and Swinnerton-Dyer, and generalizing results obtained by \textit{M. J. Razar} [Am. J. Math. 96, 104--126 (1974; Zbl 0296.14015); ibid. 127--144 (1974; Zbl 0296.14016)].