Root numbers and algebraic points on elliptic surfaces with elliptic base
The authors prove the following result on root numbers, which is independent of any conjectures. Theorem: Let \({\mathcal E}\) be an elliptic surface defined over a number field \(F\), with base an elliptic curve \(C\) and with nonconstant \(j\)-invariant. Then there exist a finite extension \(K\) of \(F\), a finite Galois extension \(L\) of \(K\), and a complex irreducible representation \(\tau\) of \(\text{Gal} (L/K)\), with real-valued character, such that, if \(E_P\) denotes the fiber of \({\mathcal E}\) over \(P\), then the set of \(P\)'s in \(C(K)\) for which the root number \(W(E_P/K,\tau)\) equals \(-1\) is infinite. Moreover, \(L\) can be chosen such that the dimension of \(\tau\) is arbitrarily large. From this theorem, it follows that if \({\mathcal E}\) has nonconstant \(j\)-invariant, then for infinitely many \(P\)'s in \(C(K)\), \(E_P(L)\) has rank at least equal to \(\dim\tau\), so \({\mathcal E}(L)\) is Zariski dense in \(E\). A similar result when the base of the elliptic surface is \(\mathbb{P}^1\) was given by the authors in Duke Math. J. 89, 413-422 (1997; Zbl 0907.14012).
scientific article; zbMATH DE number 1425125