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Jumps in Mordell-Weil rank and arithmetic surjectivity. (English) Zbl 1075.14021

Poonen, Bjorn (ed.) et al., Arithmetic of higher-dimensional algebraic varieties. Proceedings of the workshop on rational and integral points of higher-dimensional varieties, Palo Alto, CA, USA, December 11–20, 2002. Boston, MA: Birkhäuser (ISBN 0-8176-3259-X/hbk). Progress in Mathematics 226, 141-147 (2004).
Let \(f \colon X \rightarrow B\) be a smooth proper morphism over a number field \(K\). It is called arithmetically surjective if for any number field extension \(L/K\) the mapping on rational points \(f(L) \colon X(L) \rightarrow B(L)\) is surjective. Three related questions about \(f\) are posed by the authors.
Question 1: is it true that a pencil \(f \colon X \rightarrow B\) of smooth projective curves of genus one is arithmetically surjective if and only if \(f\) admits a section over \(K\)?
Then the authors restate this question as follows. Consider an open subscheme \(B\) of \( \mathbb{P}^1\) over \(K\) and the family \(X\rightarrow B\). Let \(E \rightarrow B\) be the Jacobian of this family. Then \(X \rightarrow B\) is determined up to isomorphism by some element \(h\in H^1(B,E)\) and hence also by an \( \tilde{h}\in H^1(B, E[n])\) where \(E[n]\) is the kernel of multiplication by \(n\) in \(E\), so it is natural to denote it by \(X_h \rightarrow B\) or \(X_{\tilde{h}} \rightarrow B\). In this notation Question 2 is: it is true that if \(X_h \rightarrow B\) admits no section then there exists an extension \(L/K\) and an \(L\)-valued point \(\beta \colon \text{Spec} L \rightarrow B\) such that the fiber over \(\beta\) of the family \(X_{\tilde{h}} \rightarrow B\) has no \(L\)-rational points?
Question 3 concerns the number of fibers \(X_b\) in the smooth family \(E \rightarrow B\) of elliptic curves over \(B\): is there a positive number \(e\) such that for any \(\tilde{h} \in H^1(B, E[n])\) not in the image of \(H^0(B,E)\), \[ \text{card}\{ b\in B(K) \mid \text{height}(b) < C, b^*(h)\neq 0\} > C^e \] for sufficiently large \(C\)?
Let \(K= \mathbb{Q}\), \(E_1\) be any elliptic curve over \( \mathbb{Q}\) with no points of order two rational over \( \mathbb{Q}\). Suppose that \(E \rightarrow B\) is its associated quadratic twist family. In this case the authors prove that question 3 for \(n=2\) has an affirmative answer. This is done by controlling the number of jump points for considered family.
The paper ends with an discussion on J. W. S. Cassels’ and A. Schinzel’s example [Bull. Lond. Math. Soc. 14, 345–348 (1982; Zbl 0474.14010)] of a pencil of elliptic curves over \( \mathbb{Q}\), which may be relevant for considered problems.
For the entire collection see [Zbl 1054.11006].

MSC:

14G25 Global ground fields in algebraic geometry
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G05 Rational points

Citations:

Zbl 0474.14010
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