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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
DOI: https://doi.org/10.1090/tran/7060
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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  • [BCP] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, Computational algebra and number theory (London, 1993), J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. MR 1484478, https://doi.org/10.1006/jsco.1996.0125
  • [BE] Nils Bruin and Noam D. Elkies, Trinomials $ ax^7+bx+c$ and $ ax^8+bx+c$ with Galois groups of order 168 and $ 8\cdot168$, Algorithmic number theory (Sydney, 2002) Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 172-188. MR 2041082, https://doi.org/10.1007/3-540-45455-1_14
  • [BPS16] Nils Bruin, Bjorn Poonen, and Michael Stoll, Generalized explicit descent and its application to curves of genus 3, Forum Math. Sigma 4 (2016), e6, 80. MR 3482281, https://doi.org/10.1017/fms.2016.1
  • [ES] Dennis Eriksson and Victor Scharaschkin, On the Brauer-Manin obstruction for zero-cycles on curves, Acta Arith. 135 (2008), no. 2, 99-110. MR 2453526, https://doi.org/10.4064/aa135-2-1
  • [La] Serge Lang, Abelian varieties, Springer-Verlag, New York-Berlin, 1983. Reprint of the 1959 original. MR 713430
  • [LT] Dino Lorenzini and Thomas J. Tucker, Thue equations and the method of Chabauty-Coleman, Invent. Math. 148 (2002), no. 1, 47-77. MR 1892843, https://doi.org/10.1007/s002220100186
  • [MP] William McCallum and Bjorn Poonen, The method of Chabauty and Coleman, Explicit methods in number theory, Panor. Synthèses, vol. 36, Soc. Math. France, Paris, 2012, pp. 99-117 (English, with English and French summaries). MR 3098132
  • [Mi1] J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103-150. MR 861974
  • [Mi3] J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. MR 881804
  • [Mi2] J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 167-212. MR 861976
  • [PS] Bjorn Poonen and Edward F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line, J. Reine Angew. Math. 488 (1997), 141-188. MR 1465369
  • [PSS] Bjorn Poonen, Edward F. Schaefer, and Michael Stoll, Twists of $ X(7)$ and primitive solutions to $ x^2+y^3=z^7$, Duke Math. J. 137 (2007), no. 1, 103-158. MR 2309145, https://doi.org/10.1215/S0012-7094-07-13714-1
  • [Sc1] Edward F. Schaefer, Class groups and Selmer groups, J. Number Theory 56 (1996), no. 1, 79-114. MR 1370197, https://doi.org/10.1006/jnth.1996.0006
  • [Sc2] Edward F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), no. 3, 447-471. MR 1612262, https://doi.org/10.1007/s002080050156
  • [Si] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210
  • [SvL] Michael Stoll and Ronald van Luijk, Explicit Selmer groups for cyclic covers of $ \mathbb{P}^1$, Acta Arith. 159 (2013), no. 2, 133-148. MR 3062912, https://doi.org/10.4064/aa159-2-4

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Additional Information

Edward F. Schaefer
Affiliation: Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053
Email: eschaefer@scu.edu

DOI: https://doi.org/10.1090/tran/7060
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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