Root numbers and algebraic points on elliptic surfaces with elliptic base
type: Item
instance of: scholarly article
title: Root numbers and algebraic points on elliptic surfaces with elliptic base
zbMATH Open document ID: 1029.11028
DOI: 10.1215/S0012-7094-98-09316-4
author: Gregory R. Grant
author: Elisabetta Manduchi
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: 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).
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 1425125
zbMATH Keywords: elliptic surface
zbMATH Keywords: elliptic curve
zbMATH Keywords: nonconstant \(j\)-invariant
zbMATH Keywords: root number
MaRDI profile type: Publication
cites work: Lang maps and Harris's conjecture
cites work: Q4888847
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cites work: Hyperbolic and Diophantine analysis
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description: scientific article; zbMATH DE number 1425125