Calculating root numbers of elliptic curves over \(\mathbb{Q}\). (English) Zbl 0805.14017

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\). For each prime number \(p\) and the ‘prime at infinity’ \(p = \infty\) there is associated to \(E\) the local root number \(W_ p = W_ p(E)\). These are intrinsic invariants of \(E\). For all \(p\), \(W_ p = \pm 1\), and for all but finitely many \(p\), \(W_ p=1\). The “root number” is then defined to be \(W(E) = \prod_{p \leq \infty} W_ p (E)\). This plays an important role in number theory, and because of this it is desirable to be able to write formulas for \(W\) that can be immediately translated into computer code. The present paper builds on an earlier paper of D. E. Rohrlich [Compos. Math. 87, No. 2, 119-151 (1993; Zbl 0791.11026)]. This latter paper determines \(W_ p(E)\) for all \(p \geq 5\) and all \(E\) over \(\mathbb{Q}\), and for \(p=2, 3\) and certain \(E\). Many of these determinations are quite explicit, but some of the difficult cases require some nontrivial work to put the results in a form ready for computer implementation. The present paper carries out this work. For technical reasons, it is assumed throughout that the \(j\)-invariant of \(E\) is not 0 or 1728.


14H52 Elliptic curves
11G05 Elliptic curves over global fields
14Q05 Computational aspects of algebraic curves


Zbl 0791.11026


Full Text: DOI EuDML


[1] B. J. Birch and N. M. Stephens,The parity of the rank of the Mordell-Weil group, Topology5 (1966), 295–299 · Zbl 0146.42401
[2] I. Connell,Good reduction of elliptic curves in abelian extensions, J. reine angew. Math.436 (1993), 155–175 · Zbl 0759.14025
[3] J. Cremona,Algorithms for modular elliptic curves, Cambridge University Press, London, 1992 · Zbl 0758.14042
[4] E. Liverance,A formula for the root number of a family of elliptic curves, preprint, 1992 · Zbl 0831.14012
[5] M. Kuwata and L. Wang,Topology of rational points on isotrivial elliptic surfaces, preprint, 1992 · Zbl 0804.14008
[6] D. Rohrlich,Variation of the root number in families of elliptic curves, Compos. Math. (to appear) · Zbl 0791.11026
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