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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
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