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**A formula for the root number of a family of elliptic curves.**
*(English)*
Zbl 0831.14012

This article considers the class of elliptic curves \(E_D : y^2 = x^3 + D\). Since such curves have complex multiplication, their \(L\)- functions, \(L_D (s)\), have long been known to have analytic continuation and functional equations. The sign in the functional equation (i.e., the root number) determines the parity of the order of vanishing of \(L_D (s)\) at \(s = 1\). Conjecturally, this is the same as the parity of the rank of the Mordell-Weil group \(E(\mathbb{Q})\). The author proves a closed formula for the root number of \(E_D\) as a product of simple local factors for \(p |6D\). The proof is based on the fact that \(L_D\) is a Hecke \(L\)-function, making it seem plausible that the techniques would extend to the other families of elliptic curves over \(\mathbb{Q}\) with complex multiplication. The paper also describes a known method of determining root numbers computationally.

Reviewer: J.Jones (Tempe)

### MSC:

14H52 | Elliptic curves |

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

11G05 | Elliptic curves over global fields |

14Q05 | Computational aspects of algebraic curves |