@article{09baf639dfbc40408b1c2ef6286842dd,
title = "On Mordell-Weil groups of Jacobians over function fields",
abstract = "We study the arithmetic of abelian varieties over K= k(t) where k is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over K to homomorphisms of other Jacobians over k. Our methods also yield completely explicit points on elliptic curves with unbounded rank over p(t) and a new construction of elliptic curves with moderately high rank over ℂ(t).",
keywords = "Jacobian, Mordell{Weil group, abelian variety, function field, rank",
author = "Douglas Ulmer",
note = "Funding Information: (3) A more interesting question is that of whether there are arbitrarily large d such that there exists at least one a for which the rank jumps. If so, we would have unbounded ranks over C(t). It seems likely to the author that this does not happen, in other words,thatforallsufficientlylargedandalla∈U,RankEnd(Ad,a)µd=φ(d). (4) There are at least two other families of curves with countable, dense CM subsets that could be used in conjunction with the rank formula. See [11, p. 114] Acknowledgements. It is a pleasure to acknowledge the support of several institutions in France during a sabbatical year when much of the work on this paper was carried out: IHES, the Universit{\'e}de Paris-Sud, the Universit{\'e}de Paris-VI, and the Universit{\'e}de Paris-VII. Thanks are also due to Kirti Joshi and Dinesh Thakur for many valuable conversations related to this project, to Lisa Berger and Tommy Occhipinti for their comments and questions, and to Johan de Jong and Rutger Noot for comments on their work on CM Jacobians. This research was partially supported by NSF grant DMS 0701053.",
year = "2013",
month = jan,
doi = "10.1017/S1474748012000618",
language = "English (US)",
volume = "12",
pages = "1--29",
journal = "Journal of the Institute of Mathematics of Jussieu",
issn = "1474-7480",
publisher = "Cambridge University Press",
number = "1",
}