Visualising Sha[2] in Abelian surfaces

Bruin, Nils

doi:10.1090/S0025-5718-04-01633-3

Visualising Sha[2] in Abelian surfaces
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Math. Comp. 73 (2004), 1459-1476 Request permission

Abstract:

Given an elliptic curve $E_1$ over a number field $K$ and an element $s$ in its $2$-Selmer group, we give two different ways to construct infinitely many Abelian surfaces $A$ such that the homogeneous space representing $s$ occurs as a fibre of $A$ over another elliptic curve $E_2$. We show that by comparing the $2$-Selmer groups of $E_1$, $E_2$ and $A$, we can obtain information about $\Sha (E_1/K)[2]$ and we give examples where we use this to obtain a sharp bound on the Mordell-Weil rank of an elliptic curve.

As a tool, we give a precise description of the $m$-Selmer group of an Abelian surface $A$ that is $m$-isogenous to a product of elliptic curves $E_1\times E_2$.

One of the constructions can be applied iteratively to obtain information about $\Sha (E_1/K)[2^n]$. We give an example where we use this iterated application to exhibit an element of order $4$ in $\Sha (E_1/\mathbb {Q})$.

References

Amod Agashe and William Stein, Visibility of Shafarevich-Tate groups of abelian varieties, J. Number Theory 97 (2002), no. 1, 171–185. MR 1939144, DOI 10.1006/jnth.2002.2810
M. F. Atiyah and C. T. C. Wall, Cohomology of groups, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 94–115. MR 0219512
Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822, DOI 10.1007/978-3-642-51438-8
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).
Nils Bruin, Algae, a program for $2$-Selmer groups of elliptic curves over number fields, see http://www.cecm.sfu.ca/~bruin/ell.shar.
—, Transcript of computations, available from http://www.cecm.sfu.ca/~bruin/vissha, 2002.
J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193–291. MR 199150, DOI 10.1112/jlms/s1-41.1.193
—, Lectures on elliptic curves, LMS-ST 24, University Press, Cambridge, 1991.
J. W. S. Cassels, Second descents for elliptic curves, J. Reine Angew. Math. 494 (1998), 101–127. Dedicated to Martin Kneser on the occasion of his 70th birthday. MR 1604468, DOI 10.1515/crll.1998.001
J. W. S. Cassels and E. V. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus $2$, London Mathematical Society Lecture Note Series, vol. 230, Cambridge University Press, Cambridge, 1996. MR 1406090, DOI 10.1017/CBO9780511526084
J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
J. E. Cremona, Classical invariants and 2-descent on elliptic curves, J. Symbolic Comput. 31 (2001), no. 1-2, 71–87. Computational algebra and number theory (Milwaukee, WI, 1996). MR 1806207, DOI 10.1006/jsco.1998.1004
John E. Cremona and Barry Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), no. 1, 13–28. MR 1758797
M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, KANT V4, J. Symbolic Comput. 24 (1997), no. 3-4, 267–283. Computational algebra and number theory (London, 1993). MR 1484479, DOI 10.1006/jsco.1996.0126
Z. Djabri, Edward F. Schaefer, and N. P. Smart, Computing the $p$-Selmer group of an elliptic curve, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5583–5597. MR 1694286, DOI 10.1090/S0002-9947-00-02535-6
E. V. Flynn and J. Redmond, Application of covering techniques to families of curves, J. Number Theory 101 (2003), no. 2, 376–397.
Florian Heß, Zur Klassengruppenberechnung in algebraischen Zahlkörpern, Diplomarbeit, Technische Universität Berlin, 1996, available from http://www.math.tu-berlin. de/~kant/publications/diplom/hess.ps.gz.
Tomas Antonius Klenke, Visualizing elements of order two in the Weil-Châtelet group, in preparation, 2001.
Kenneth Kramer, Arithmetic of elliptic curves upon quadratic extension, Trans. Amer. Math. Soc. 264 (1981), no. 1, 121–135. MR 597871, DOI 10.1090/S0002-9947-1981-0597871-8
J. R. Merriman, S. Siksek, and N. P. Smart, Explicit $4$-descents on an elliptic curve, Acta Arith. 77 (1996), no. 4, 385–404. MR 1414518, DOI 10.4064/aa-77-4-385-404
Edward F. Schaefer and Michael Stoll, How to do a $p$-descent on an elliptic curve, see http://www.math.uni-duesseldorf.de/~stoll/papers/p-d escent.dvi.
S. Siksek and N. P. Smart, On the complexity of computing the $2$-Selmer group of an elliptic curve, Glasgow Math. J. 39 (1997), no. 3, 251–257. MR 1484568, DOI 10.1017/S0017089500032183
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Denis Simon, Computing the rank of elliptic curves over number fields, LMS J. Comput. Math. 5 (2002), 7–17. MR 1916919, DOI 10.1112/S1461157000000668
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