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Visualizing elements of order $7$ in the Tate–Shafarevich group of an elliptic curve

Published online by Cambridge University Press:  26 August 2016

Tom Fisher*
Affiliation:
University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom email T.A.Fisher@dpmms.cam.ac.uk

Abstract

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We study the elliptic curves in Cremona’s tables that are predicted by the Birch–Swinnerton-Dyer conjecture to have elements of order $7$ in their Tate–Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian 3-fold.

Type
Research Article
Copyright
© The Author 2016 

References

Adler, A. and Ramanan, S., Moduli of abelian varieties , Lecture Notes in Mathematics 1644 (Springer, Berlin, 1996).CrossRefGoogle Scholar
Agashé, A. and Stein, W. A., ‘Visibility of Shafarevich–Tate groups of abelian varieties’, J. Number Theory 97 (2002) no. 1, 171185.CrossRefGoogle Scholar
Agashé, A. and Stein, W. A., ‘Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, with an appendix by J. E. Cremona and B. Mazur’, Math. Comp. 74 (2005) no. 249, 455484.CrossRefGoogle Scholar
An, S. Y., Kim, S. Y., Marshall, D. C., Marshall, S. H., McCallum, W. G. and Perlis, A. R., ‘Jacobians of genus one curves’, J. Number Theory 90 (2001) no. 2, 304315.CrossRefGoogle Scholar
Bending, P. R., ‘Curves of genus $2$ with $\sqrt{2}$ multiplication’, Preprint, 1999, arXiv:9911273.Google Scholar
Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 21 (Springer, Berlin, 1990).CrossRefGoogle Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: The user language’, J. Symbolic Comput. 24 (1997) 235265, see also http://magma.maths.usyd.edu.au/magma/.CrossRefGoogle Scholar
Cassels, J. W. S. and Flynn, E. V., Prolegomena to a middlebrow arithmetic of curves of genus 2 , LMS Lecture Note Series 230 (Cambridge University Press, Cambridge, 1996).CrossRefGoogle Scholar
Cremona, J. E., Algorithms for modular elliptic curves (Cambridge University Press, Cambridge, 1997) See also http://www.warwick.ac.uk/∼masgaj/ftp/data/.Google Scholar
Cremona, J. E. and Mazur, B., ‘Visualizing elements in the Shafarevich–Tate group’, Exp. Math. 9 (2000) no. 1, 1328.CrossRefGoogle Scholar
Fisher, T. A., ‘Finding rational points on elliptic curves using 6-descent and 12-descent’, J. Algebra 320 (2008) no. 2, 853884.CrossRefGoogle Scholar
Fisher, T. A., ‘On families of 7 and 11-congruent elliptic curves’, LMS J. Comput. Math. 17 (2014) no. 1, 536564.CrossRefGoogle Scholar
Fisher, T. A., ‘Invisibility of Tate–Shafarevich groups in abelian surfaces’, Int. Math. Res. Not. IMRN 2014 no. 15, 40854099.CrossRefGoogle Scholar
Flynn, E. V., Leprévost, F., Schaefer, E. F., Stein, W. A., Stoll, M. and Wetherell, J. L., ‘Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves’, Math. Comp. 70 (2001) no. 236, 16751697.CrossRefGoogle Scholar
González-Jiménez, E., González, J. and Guàrdia, J., ‘Computations on modular Jacobian surfaces’, Algorithmic number theory (Sydney, 2002) , Lecture Notes in Computational Science 2369 (Springer, Berlin, 2002) 189197.CrossRefGoogle Scholar
Grigorov, G., Jorza, A., Patrikis, S., Stein, W. A. and Tarnita, C., ‘Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves’, Math. Comp. 78 (2009) no. 268, 23972425.CrossRefGoogle Scholar
Halberstadt, E. and Kraus, A., ‘Sur la courbe modulaire X E (7)’, Exp. Math. 12 (2003) no. 1, 2740.CrossRefGoogle Scholar
Kraus, A. and Oesterlé, J., ‘Sur une question de B. Mazur’, Math. Ann. 293 (1992) no. 2, 259275.CrossRefGoogle Scholar
Mazur, B., ‘Visualizing elements of order three in the Shafarevich–Tate group’, Asian J. Math. 3 (1999) no. 1, 221232.CrossRefGoogle Scholar
Merriman, J. R. and Smart, N. P., ‘Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point’, Math. Proc. Cambridge Philos. Soc. 114 (1993) no. 2, 203214.CrossRefGoogle Scholar
PARI/GP, Version 2.5.5, Bordeaux, 2013, http://pari.math.u-bordeaux.fr/.Google Scholar
Poonen, B., Schaefer, E. F. and Stoll, M., ‘Twists of X (7) and primitive solutions to x 2 + y 3 = z 7 ’, Duke Math. J. 137 (2007) no. 1, 103158.CrossRefGoogle Scholar
Schaefer, E. F., ‘Class groups and Selmer groups’, J. Number Theory 56 (1996) no. 1, 79114.CrossRefGoogle Scholar
Serre, J.-P., ‘Propriétés galoisiennes des points d’ordre fini des courbes elliptiques’, Invent. Math. 15 (1972) no. 4, 259331.CrossRefGoogle Scholar
Shimura, G., ‘On the factors of the Jacobian variety of a modular function field’, J. Math. Soc. Japan 25 (1973) 523544.CrossRefGoogle Scholar
Siksek, S., ‘Infinite descent on elliptic curves’, Rocky Mountain J. Math. 25 (1995) no. 4, 15011538.CrossRefGoogle Scholar
Silverman, J. H., ‘The Néron fiber of abelian varieties with potential good reduction’, Math. Ann. 264 (1983) no. 1, 13.CrossRefGoogle Scholar
Silverman, J. H., The arithmetic of elliptic curves , Graduate Text in Mathematics 106 (Springer, New York, 1986).CrossRefGoogle Scholar
Stein, W. A. et al. , ‘Sage Mathematics Software (Version 6.2)’, The Sage Development Team, 2014, http://www.sagemath.org.Google Scholar
Stoll, M., ‘Two simple 2-dimensional abelian varieties defined over ℚ with Mordell–Weil group of rank at least 19’, C. R. Math. Acad. Sci. Paris 321 (1995) no. I, 13411345.Google Scholar
Vélu, J., ‘Isogénies entre courbes elliptiques’, C. R. Math. Acad. Sci. Paris Sér. A-B 273 (1971) A238A241.Google Scholar
van Wamelen, P. B., ‘Computing with the analytic Jacobian of a genus 2 curve’, Discovering mathematics with Magma , Algorithms and Computation in Mathematics 19 (eds Bosma, W. and Cannon, J.; Springer, Berlin, 2006) 117135.CrossRefGoogle Scholar
Wang, X. D., ‘2-dimensional simple factors of J 0(N)’, Manuscripta Math. 87 (1995) no. 2, 179197.CrossRefGoogle Scholar