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Variation of the root number in families of elliptic curves. (English) Zbl 0791.11026
Let \(X\) be an elliptic surface over \(\mathbb{A}^ 1(\mathbb{Q})\). For \(t\in \mathbb{Q}\) let \(E_ t\) denote the fiber over \(t\). The author investigates the sets \(T^ \pm := \{t \in \mathbb{Q} \mid W(E_ t) = \pm 1\}\); here \(W(E)\) denotes the root number of \(E\). For example, the paper gives examples where there exists a polynomial \(f(t)\) such that \(t \in T^ +\) if and only if \(f(t) > 0\). This project was inspired by some recent ideas of B. Mazur [Exp. Math. 1, 35-45 (1992; Zbl 0784.14012)].
Reviewer: P.Vojta (Berkeley)

MSC:
11G05 Elliptic curves over global fields
14G05 Rational points
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