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Explicit descent for Jacobians of prime power cyclic covers of the projective line

Author: Edward F. Schaefer
Journal: Trans. Amer. Math. Soc. 370 (2018), 3487-3505
MSC (2010): Primary 11G30; Secondary 11G10, 14G25, 14H40, 14H45
Published electronically: December 1, 2017
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Abstract: The Jacobian of a cyclic cover of the projective line is isogenous to a product of abelian subvarieties, one for each positive divisor of the degree of the cover. In this article, we show how to compute a Selmer group that bounds the Mordell-Weil rank for each abelian subvariety corresponding to a non-trivial prime power divisor of the degree. In the case that the Chabauty condition holds for that abelian subvariety, we show how to bound the number of rational points on the curve.

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Additional Information

Edward F. Schaefer
Affiliation: Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053

Received by editor(s): July 29, 2015
Received by editor(s) in revised form: August 10, 2016
Published electronically: December 1, 2017
Additional Notes: The author is grateful for the hospitality of the Mathematisches Institut at the Universität Bayreuth, where much of this research was done, and to his host there, Michael Stoll, for many useful conversations. This article benefited from useful comments from the referees.
Article copyright: © Copyright 2017 American Mathematical Society

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