Computing integral points on hyperelliptic curves using quadratic Chabauty

Balakrishnan, Jennifer; Besser, Amnon; Müller, J.

doi:10.1090/mcom/3130

Computing integral points on hyperelliptic curves using quadratic Chabauty
HTML articles powered by AMS MathViewer

by Jennifer S. Balakrishnan, Amnon Besser and J. Steffen Müller;

Math. Comp. 86 (2017), 1403-1434

DOI: https://doi.org/10.1090/mcom/3130

Published electronically: October 12, 2016

PDF | Request permission

Abstract:

We give a method for the computation of integral points on a hyperelliptic curve of odd degree over the rationals whose genus equals the Mordell-Weil rank of its Jacobian. Our approach consists of a combination of the $p$-adic approximation techniques introduced in previous work with the Mordell-Weil sieve.

References

Matthew H. Baker, Enrique González-Jiménez, Josep González, and Bjorn Poonen, Finiteness results for modular curves of genus at least 2, Amer. J. Math. 127 (2005), no. 6, 1325–1387. MR 2183527
Jennifer S. Balakrishnan, Iterated Coleman integration for hyperelliptic curves, ANTS X—Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Ser., vol. 1, Math. Sci. Publ., Berkeley, CA, 2013, pp. 41–61. MR 3207407, DOI 10.2140/obs.2013.1.41
Jennifer S. Balakrishnan and Amnon Besser, Computing local $p$-adic height pairings on hyperelliptic curves, Int. Math. Res. Not. IMRN 11 (2012), 2405–2444. MR 2926986, DOI 10.1093/imrn/rnr111
Jennifer S. Balakrishnan and Amnon Besser, Coleman-Gross height pairings and the $p$-adic sigma function, J. Reine Angew. Math. 698 (2015), 89–104. MR 3294652, DOI 10.1515/crelle-2012-0095
J. S. Balakrishnan, A. Besser, and J. S. Müller, Quadratic Chabauty: $p$-adic heights and integral points on hyperelliptic curves, J. Reine Angew. Math (2015), to appear.
Jennifer S. Balakrishnan, Robert W. Bradshaw, and Kiran S. Kedlaya, Explicit Coleman integration for hyperelliptic curves, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 6197, Springer, Berlin, 2010, pp. 16–31. MR 2721410, DOI 10.1007/978-3-642-14518-6_{6}
J. S. Balakrishnan, I. Dan-Cohen, M. Kim, and S. Wewers, A non-abelian conjecture of Birch and Swinnerton-Dyer type for hyperbolic curves, preprint (2014), arXiv:1209.0640.
Jennifer S. Balakrishnan, Kiran S. Kedlaya, and Minhyong Kim, Appendix and erratum to “Massey products for elliptic curves of rank 1” [MR2629986], J. Amer. Math. Soc. 24 (2011), no. 1, 281–291. MR 2726605, DOI 10.1090/S0894-0347-2010-00675-3
Jennifer S. Balakrishnan, J. Steffen Müller, and William A. Stein, A $p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties, Math. Comp. 85 (2016), no. 298, 983–1016. MR 3434891, DOI 10.1090/mcom/3029
Amnon Besser, $p$-adic Arakelov theory, J. Number Theory 111 (2005), no. 2, 318–371. MR 2130113, DOI 10.1016/j.jnt.2004.11.010
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Nils Bruin, Chabauty methods using elliptic curves, J. Reine Angew. Math. 562 (2003), 27–49. MR 2011330, DOI 10.1515/crll.2003.076
Nils Bruin and Noam D. Elkies, Trinomials $ax^7+bx+c$ and $ax^8+bx+c$ with Galois groups of order 168 and $8\cdot 168$, Algorithmic number theory (Sydney, 2002) Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 172–188. MR 2041082, DOI 10.1007/3-540-45455-1_{1}4
Nils Bruin and Michael Stoll, Deciding existence of rational points on curves: an experiment, Experiment. Math. 17 (2008), no. 2, 181–189. MR 2433884
Nils Bruin and Michael Stoll, Two-cover descent on hyperelliptic curves, Math. Comp. 78 (2009), no. 268, 2347–2370. MR 2521292, DOI 10.1090/S0025-5718-09-02255-8
Nils Bruin and Michael Stoll, The Mordell-Weil sieve: proving non-existence of rational points on curves, LMS J. Comput. Math. 13 (2010), 272–306. MR 2685127, DOI 10.1112/S1461157009000187
Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll, and Szabolcs Tengely, Integral points on hyperelliptic curves, Algebra Number Theory 2 (2008), no. 8, 859–885. MR 2457355, DOI 10.2140/ant.2008.2.859
Claude Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885 (French). MR 4484
Robert F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770. MR 808103, DOI 10.1215/S0012-7094-85-05240-8
Robert F. Coleman and Benedict H. Gross, $p$-adic heights on curves, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 73–81. MR 1097610, DOI 10.2969/aspm/01710073
Sinnou David, Minorations de formes linéaires de logarithmes elliptiques, Mém. Soc. Math. France (N.S.) 62 (1995), iv+143 (French, with English and French summaries). MR 1385175
E. Victor Flynn, Coverings of curves of genus 2, Algorithmic number theory (Leiden, 2000) Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, pp. 65–84. MR 1850599, DOI 10.1007/10722028_{3}
E. V. Flynn, The Hasse principle and the Brauer-Manin obstruction for curves, Manuscripta Math. 115 (2004), no. 4, 437–466. MR 2103661, DOI 10.1007/s00229-004-0502-9
E. V. Flynn and N. P. Smart, Canonical heights on the Jacobians of curves of genus $2$ and the infinite descent, Acta Arith. 79 (1997), no. 4, 333–352. MR 1450916, DOI 10.4064/aa-79-4-333-352
E. Victor Flynn and Joseph L. Wetherell, Covering collections and a challenge problem of Serre, Acta Arith. 98 (2001), no. 2, 197–205. MR 1831612, DOI 10.4064/aa98-2-9
Benedict H. Gross, Local heights on curves, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 327–339. MR 861983
N. Hirata-Kohno and T. Kovacs, Computing $S$-integral points on elliptic curves of rank at least 3, RIMS Kokyuroku 1898 (2014), 92–102.
Minhyong Kim, The motivic fundamental group of $\mathbf P^1\sbs \{0,1,\infty \}$ and the theorem of Siegel, Invent. Math. 161 (2005), no. 3, 629–656. MR 2181717, DOI 10.1007/s00222-004-0433-9
Minhyong Kim, Massey products for elliptic curves of rank 1, J. Amer. Math. Soc. 23 (2010), no. 3, 725–747. MR 2629986, DOI 10.1090/S0894-0347-10-00665-X
Serge Lang, Introduction to Arakelov theory, Springer-Verlag, New York, 1988. MR 969124, DOI 10.1007/978-1-4612-1031-3
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
J. Steffen Müller and Michael Stoll, Canonical heights on genus two Jacobians, in preparation, 2015.
R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406–413. MR 69793
Attila Pethő, Horst G. Zimmer, Josef Gebel, and Emanuel Herrmann, Computing all $S$-integral points on elliptic curves, Math. Proc. Cambridge Philos. Soc. 127 (1999), no. 3, 383–402. MR 1713117, DOI 10.1017/S0305004199003916
Stephen C. Pohlig and Martin E. Hellman, An improved algorithm for computing logarithms over $\textrm {GF}(p)$ and its cryptographic significance, IEEE Trans. Inform. Theory IT-24 (1978), no. 1, 106–110. MR 484737, DOI 10.1109/tit.1978.1055817
Bjorn Poonen, Heuristics for the Brauer-Manin obstruction for curves, Experiment. Math. 15 (2006), no. 4, 415–420. MR 2293593
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, DOI 10.1215/S0012-7094-07-13714-1
Victor Scharaschkin, Local-global problems and the Brauer-Manin obstruction, ProQuest LLC, Ann Arbor, MI, 1999. Thesis (Ph.D.)–University of Michigan. MR 2700328
Joseph H. Silverman and Katherine E. Stange, Amicable pairs and aliquot cycles for elliptic curves, Exp. Math. 20 (2011), no. 3, 329–357. MR 2836257, DOI 10.1080/10586458.2011.565253
N. P. Smart, $S$-integral points on elliptic curves, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 3, 391–399. MR 1291748, DOI 10.1017/S0305004100072698
Nigel P. Smart, The algorithmic resolution of Diophantine equations, London Mathematical Society Student Texts, vol. 41, Cambridge University Press, Cambridge, 1998. MR 1689189, DOI 10.1017/CBO9781107359994
W. A. Stein et al., Sage Mathematics Software (Version 6.5), The Sage Development Team, 2015, http://www.sagemath.org.
Michael Stoll, On the height constant for curves of genus two, Acta Arith. 90 (1999), no. 2, 183–201. MR 1709054, DOI 10.4064/aa-90-2-183-201
Michael Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), no. 3, 245–277. MR 1829626, DOI 10.4064/aa98-3-4
Michael Stoll, On the height constant for curves of genus two. II, Acta Arith. 104 (2002), no. 2, 165–182. MR 1914251, DOI 10.4064/aa104-2-6
M. Stoll, An explicit theory of heights for hyperelliptic jacobians of genus three, preprint (2014), http://www.mathe2.uni-bayreuth.de/stoll/papers/Kummer-g3-hyp-2014-05-15.pdf.
R. J. Stroeker and N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67 (1994), no. 2, 177–196. MR 1291875, DOI 10.4064/aa-67-2-177-196