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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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