Local solubility and height bounds for coverings of elliptic curves
Authors:
T. A. Fisher and G. F. Sills
Journal:
Math. Comp. 81 (2012), 16351662
MSC (2010):
Primary 11G05; Secondary 11G07, 11G50, 11Y50
Published electronically:
February 21, 2012
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Abstract: We study genus one curves that arise as ,  and coverings of elliptic curves. We describe efficient algorithms for testing local solubility and modify the classical formulae for the covering maps so that they work in all characteristics. These ingredients are then combined to give explicit bounds relating the height of a rational point on one of the covering curves to the height of its image on the elliptic curve. We use our results to improve the existing methods for searching for rational points on elliptic curves.
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(2009m:11078), http://dx.doi.org/10.1007/9783540794561_8
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T.A. Fisher, The Hessian of a genus one curve, to appear in Proc. Lond. Math. Soc.
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T.A. Fisher, Higher descents on an elliptic curve with a rational torsion point, in preparation.
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Marc
Hindry and Joseph
H. Silverman, Diophantine geometry, Graduate Texts in
Mathematics, vol. 201, SpringerVerlag, New York, 2000. An
introduction. MR
1745599 (2001e:11058)
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J.
R. Merriman, S.
Siksek, and N.
P. Smart, Explicit 4descents on an elliptic curve, Acta
Arith. 77 (1996), no. 4, 385–404. MR 1414518
(97j:11027)
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M.M. Sadek, Models of genus one curves, PhD thesis, University of Cambridge, 2009.
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Edward
F. Schaefer and Michael
Stoll, How to do a 𝑝descent on an
elliptic curve, Trans. Amer. Math. Soc.
356 (2004), no. 3,
1209–1231. MR 2021618
(2004g:11045), http://dx.doi.org/10.1090/S000299470303366X
 [Si1]
S. Siksek, Descent on curves of genus one, PhD thesis, University of Exeter, 1995.
http://www.warwick.ac.uk/staff/S.Siksek/papers/phdnew.pdf
 [Si2]
Samir
Siksek, Infinite descent on elliptic curves, Rocky Mountain J.
Math. 25 (1995), no. 4, 1501–1538. MR 1371352
(97g:11053), http://dx.doi.org/10.1216/rmjm/1181072159
 [S]
G.F. Sills, Height bounds for coverings, PhD thesis, University of Cambridge, 2010.
 [Sta]
S. Stamminger, Explicit 8descent on elliptic curves, PhD thesis, International University Bremen, 2005.
 [Sto]
M. Stoll, Descent on elliptic curves, lecture notes, arXiv:math/0611694v1 [math.NT]
 [W]
T. Womack, Explicit descent on elliptic curves, PhD thesis, University of Nottingham, 2003. http://www.warwick.ac.uk/staff/J.E.Cremona/
 [AS]
 A. Agashe and W. Stein, Visibility of ShafarevichTate groups of abelian varieties, J. Number Theory 97 (2002), no. 1, 171185. MR 1939144 (2003h:11070)
 [AKMP]
 S.Y. An, S.Y. Kim, D.C. Marshall, S.H. Marshall, W.G. McCallum and A.R. Perlis, Jacobians of genus one curves, J. Number Theory 90 (2001), no. 2, 304315. MR 1858080 (2002g:14040)
 [ARVT]
 M. Artin, F. RodriguezVillegas and J. Tate, On the Jacobians of plane cubics, Adv. Math. 198 (2005), no. 1, 366382. MR 2183258 (2006h:14043)
 [BSD]
 B.J. Birch and H.P.F. SwinnertonDyer, Notes on elliptic curves I. J. Reine Angew. Math. 212 1963 725. MR 0146143 (26:3669)
 [BCP]
 W. Bosma, J. Cannon and C. Playoust, The MAGMA algebra system I: The user language, J. Symb. Comb. 24, (1997) 235265. (See also the MAGMA home page at http://magma.maths.usyd.edu.au/magma/.) MR 1484478
 [B]
 N. Bruin, Some ternary Diophantine equations of signature , Discovering mathematics with Magma, 6391, Algorithms Comput. Math., 19, Springer, Berlin, 2006. MR 2278923 (2007m:11047)
 [Ca]
 J.W.S. Cassels, Arithmetic on curves of genus , IV. Proof of the Hauptvermutung, J. Reine Angew. Math. 211 1962 95112. MR 0163915 (29:1214)
 [Cr]
 J.E. Cremona, Algorithms for modular elliptic curves, Second edition, Cambridge University Press, Cambridge, 1997. (See also the tables at http://www.warwick.ac.uk/staff/J.E.Cremona/ftp/data/.) MR 1628193 (99e:11068)
 [Cre]
 B.M. Creutz, Explicit second descent on elliptic curves, PhD thesis, Jacobs University Bremen, 2010.
 [CFOSS]
 J.E. Cremona, T.A. Fisher, C. O'Neil, D. Simon and M. Stoll, Explicit descent on elliptic curves, I, Algebra J. Reine Angew. Math. 615 (2008) 121155; II Geometry J. Reine Angew. Math. 632 (2009), 6384; III Algorithms, preprint, arXiv:1107.3516v1 [math.NT]. MR 2384334 (2009g:11067); MR 254413 (2011d:11128)
 [CFS]
 J.E. Cremona, T.A. Fisher and M. Stoll, Minimisation and reduction of 2, 3 and 4coverings of elliptic curves, Algebra & Number Theory 4 (2010), no. 6, 763820. MR 2728489
 [CPS]
 J.E. Cremona, M. Prickett and S. Siksek, Height difference bounds for elliptic curves over number fields, J. Number Theory 116 (2006), no. 1, 4268. MR 2197860 (2006k:11121)
 [D]
 S. Donnelly, Computing the CasselsTate pairing, in preparation.
 [E]
 N.D. Elkies, Rational points near curves and small nonzero via lattice reduction, Algorithmic number theory (Leiden, 2000), 3363, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000. MR 1850598 (2002g:11035)
 [F1]
 T.A. Fisher, The invariants of a genus one curve, Proc. Lond. Math. Soc. (3) 97 (2008), no. 3, 753782. MR 2448246 (2009j:11087)
 [F2]
 T.A. Fisher, Some improvements to 4descent on an elliptic curve, in Algorithmic number theory, A. van der Poorten, A. Stein (eds.), Lecture Notes in Comput. Sci., 5011, Springer, 2008. MR 2467841 (2009m:11078)
 [F3]
 T.A. Fisher, The Hessian of a genus one curve, to appear in Proc. Lond. Math. Soc.
 [F4]
 T.A. Fisher, Higher descents on an elliptic curve with a rational torsion point, in preparation.
 [FS]
 T.A. Fisher and G.F. Sills, Local solubility and height bounds for coverings of elliptic curves, longer version of this paper, arXiv:1103.4944v1 [math.NT]
 [HS]
 M. Hindry and J.H. Silverman, Diophantine geometry, An introduction, Graduate Texts in Mathematics, 201, SpringerVerlag, New York, 2000. MR 1745599 (2001e:11058)
 [MSS]
 J. R. Merriman, S. Siksek and N. P. Smart, Explicit descents on an elliptic curve, Acta Arith. 77 (1996), no. 4, 385404. MR 1414518 (97j:11027)
 [Sa]
 M.M. Sadek, Models of genus one curves, PhD thesis, University of Cambridge, 2009.
 [SS]
 E.F. Schaefer and M. Stoll, How to do a descent on an elliptic curve, Trans. Amer. Math. Soc. 356 no. 3 (2004), 12091231 MR 2021618 (2004g:11045)
 [Si1]
 S. Siksek, Descent on curves of genus one, PhD thesis, University of Exeter, 1995.
http://www.warwick.ac.uk/staff/S.Siksek/papers/phdnew.pdf
 [Si2]
 S. Siksek, Infinite descent on elliptic curves, Rocky Mountain J. Math. 25 (1995), no. 4, 15011538. MR 1371352 (97g:11053)
 [S]
 G.F. Sills, Height bounds for coverings, PhD thesis, University of Cambridge, 2010.
 [Sta]
 S. Stamminger, Explicit 8descent on elliptic curves, PhD thesis, International University Bremen, 2005.
 [Sto]
 M. Stoll, Descent on elliptic curves, lecture notes, arXiv:math/0611694v1 [math.NT]
 [W]
 T. Womack, Explicit descent on elliptic curves, PhD thesis, University of Nottingham, 2003. http://www.warwick.ac.uk/staff/J.E.Cremona/
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Additional Information
T. A. 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
G. F. Sills
Affiliation:
University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email:
gs300@cantab.net
DOI:
http://dx.doi.org/10.1090/S002557182012025877
PII:
S 00255718(2012)025877
Received by editor(s):
October 21, 2010
Received by editor(s) in revised form:
March 28, 2011
Published electronically:
February 21, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
