Visualising Sha[2] in Abelian surfaces

Author:
Nils Bruin

Translated by:

Journal:
Math. Comp. **73** (2004), 1459-1476

MSC (2000):
Primary 11G05; Secondary 14G05, 14K15

DOI:
https://doi.org/10.1090/S0025-5718-04-01633-3

Published electronically:
January 8, 2004

MathSciNet review:
2047096

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given an elliptic curve over a number field and an element in its -Selmer group, we give two different ways to construct infinitely many Abelian surfaces such that the homogeneous space representing occurs as a fibre of over another elliptic curve . We show that by comparing the -Selmer groups of , and , we can obtain information about 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 -Selmer group of an Abelian surface that is -isogenous to a product of elliptic curves .

One of the constructions can be applied iteratively to obtain information about . We give an example where we use this iterated application to exhibit an element of order in .

**1.**Amod Agashe and William A. Stein,*Visibility of Shafarevich-Tate groups of abelian varieties*,`http://modular.fas.harvard.edu/papers/`, 2001. MR**2003h:11070****2.**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**36:2593****3.**Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud,*Néron models*, Springer-Verlag, Berlin, 1990. MR**91i:14034****4.**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).**5.**Nils Bruin,*Algae, a program for -Selmer groups of elliptic curves over number fields*, see`http://www.cecm.sfu.ca/bruin/ell.shar`.**6.**-,*Transcript of computations*, available from`http://www.cecm.sfu.ca/bruin/vissha`, 2002.**7.**J. W. S. Cassels,*Diophantine equations with special reference to elliptic curves*, J. London Math. Soc.**41**(1966), 193-291. MR**33:7299****8.**-,*Lectures on elliptic curves*, LMS-ST 24, University Press, Cambridge, 1991.**9.**-,*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**99d:11058****10.**J. W. S. Cassels and E. V. Flynn,*Prolegomena to a middlebrow arithmetic of curves of genus*, LMS-LNS 230, Cambridge University Press, Cambridge, 1996. MR**97i:11071****11.**J. E. Cremona,*Algorithms for modular elliptic curves*, Cambridge University Press, 1992. MR**93m:11053****12.**-,*Classical invariants and -descent on elliptic curves*, J. Symbolic Comput.**31**(2001), no. 1-2, 71-87, Computational algebra and number theory (Milwaukee, WI, 1996). MR**2002a:11055****13.**John E. Cremona and Barry Mazur,*Visualizing elements in the Shafarevich-Tate group*, Experiment. Math.**9**(2000), no. 1, 13-28. MR**2001g:11083****14.**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, Available from`ftp://ftp.math.tu-berlin.de/pub/algebra/Kant/Kash`. MR**99g:11150****15.**Z. Djabri, Edward F. Schaefer, and N. P. Smart,*Computing the -Selmer group of an elliptic curve*, Trans. Amer. Math. Soc.**352**(2000), no. 12, 5583-5597. MR**2001b:11047****16.**E. V. Flynn and J. Redmond,*Application of covering techniques to families of curves*, J. Number Theory**101**(2003), no. 2, 376-397.**17.**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`.**18.**Tomas Antonius Klenke,*Visualizing elements of order two in the Weil-Châtelet group*, in preparation, 2001.**19.**Kenneth Kramer,*Arithmetic of elliptic curves upon quadratic extension*, Trans. Amer. Math. Soc.**264**(1981), no. 1, 121-135. MR**82g:14028****20.**J. R. Merriman, S. Siksek, and N. P. Smart,*Explicit -descents on an elliptic curve*, Acta Arith.**77**(1996), no. 4, 385-404. MR**97j:11027****21.**Edward F. Schaefer and Michael Stoll,*How to do a -descent on an elliptic curve*, see`http://www.math.uni-duesseldorf.de/stoll/papers/p-d escent.dvi`.**22.**S. Siksek and N. P. Smart,*On the complexity of computing the -Selmer group of an elliptic curve*, Glasgow Math. J.**39**(1997), no. 3, 251-257. MR**99b:11061****23.**Joseph H. Silverman,*The arithmetic of elliptic curves*, GTM 106, Springer-Verlag, 1986. MR**87g:11070****24.**Denis Simon,*Computing the ranks of elliptic curves over number fields*, to appear in LMS JCM. MR**2003g:11060****25.**Michael Stoll,*Implementing -descent for Jacobians of hyperelliptic curves*, Acta Arith.**98**(2001), no. 3, 245-277. MR**2002b:11089**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
11G05,
14G05,
14K15

Retrieve articles in all journals with MSC (2000): 11G05, 14G05, 14K15

Additional Information

**Nils Bruin**

Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Email:
bruin@member.ams.org

DOI:
https://doi.org/10.1090/S0025-5718-04-01633-3

Keywords:
Visualisation,
Shafarevich-Tate group,
elliptic curve,
two-descent,
Mordell-Weil group

Received by editor(s):
February 2, 2002

Received by editor(s) in revised form:
September 13, 2002

Published electronically:
January 8, 2004

Additional Notes:
The research in this paper was funded by the Pacific Institute for the Mathematical Sciences, Simon Fraser University, the University of British Columbia, and the School of Mathematics of the University of Sydney

Article copyright:
© Copyright 2004
American Mathematical Society