Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-29T22:29:08.542Z Has data issue: false hasContentIssue false
  • English
  • Français

Valuations of Somos 4 sequences and canonical local heights on elliptic curves

Published online by Cambridge University Press:  12 January 2011

YUKIHIRO UCHIDA
YUKIHIRO UCHIDA*
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Somos 4 sequences are sequences of numbers defined by a bilinear recurrence relation of order 4 and include elliptic divisibility sequences as a special case. In this paper, we describe valuations of Somos 4 sequences in terms of canonical local heights on the associated elliptic curves. We consider both Archimedean and non-Archimedean valuations. As applications, we study the asymptotic behaviour of valuations of Somos 4 sequences and obtain another proof of the integrality of certain Somos 4 sequences.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 3 , May 2011 , pp. 385 - 397
Copyright
Copyright © Cambridge Philosophical Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Akiyama, S.A criterion to estimate the least common multiple of sequences and asymptotic formulas for ζ(3) arising from recurrence relation of an elliptic function. Japan. J. Math. (N.S.) 22 (1996), 129–146.CrossRefGoogle Scholar
[2]Cheon, J. and Hahn, S.Explicit valuations of division polynomials of an elliptic curve. Manuscripta Math. 97 (1998), 319–328.CrossRefGoogle Scholar
[3]Cremona, J. E., Prickett, M. and Siksek, S.Height difference bounds for elliptic curves over number fields. J. Number Theory 116 (2006), 42–68.CrossRefGoogle Scholar
[4]David, S. and Hirata-Kohno, N.Linear forms in elliptic logarithms. J. Reine Angew. Math. 628 (2009), 37–89.Google Scholar
[5]Everest, G. and Ward, T.The canonical height of an algebraic point on an elliptic curve. New York J. Math. 6 (2000), 331–342.Google Scholar
[6]Fomin, S. and Zelevinsky, A.The Laurent phenomenon. Adv. in Appl. Math. 28 (2002), 119–144.CrossRefGoogle Scholar
[7]Hirata–Kohno, N. and Takada, R.Linear forms in two elliptic logarithms in the p-adic case. Kyushu J. Math. 64 (2010), 239–260.CrossRefGoogle Scholar
[8]Hone, A. N. W.Sigma function solution of the initial value problem for Somos 5 sequences. Trans. Amer. Math. Soc. 359 (2007), 5019–5034.CrossRefGoogle Scholar
[9]Hone, A. N. W. and Swart, C.Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences. Math. Proc. Camb. Phil. Soc. 145 (2008), 65–85.CrossRefGoogle Scholar
[10]Lutz, E.Sur l'équation y 2 = x 3 − Ax − B dans les corps -adiques. J. Reine Angew. Math. 177 (1937), 238–247.CrossRefGoogle Scholar
[11]Malouf, J. L.An integer sequence from a rational recursion. Discrete Math. 110 (1992), 257–261.CrossRefGoogle Scholar
[12]van der Poorten, A. J. and Swart, C. S.Recurrence relations for elliptic sequences: every Somos 4 is a Somos k. Bull. London Math. Soc. 38 (2006), 546–554.CrossRefGoogle Scholar
[13]Rémond, G. and Urfels, F.Approximation diophantienne de logarithmes elliptiques p-adiques. J. Number Theory 57 (1996), 133–169.CrossRefGoogle Scholar
[14]Schmitt, S. and Zimmer, H. G.Elliptic Curves: A Computational Approach. de Gruyter Studies in Mathematics 31 (Walter de Gruyter, 2003).Google Scholar
[15]Silverman, J. H.The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics vol. 106 (Springer-Verlag, 1986).CrossRefGoogle Scholar
[16]Silverman, J. H.Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics vol. 151 (Springer-Verlag, 1994).CrossRefGoogle Scholar
[17]Swart, C. S. Elliptic curves and related sequences. PhD. thesis. Royal Holloway, University of London (2003); available at http://www.isg.rhul.ac.uk/files/alumni/thesis/swart_c.pdf.Google Scholar
[18]Uchida, Y.The difference between the ordinary height and the canonical height on elliptic curves. J. Number Theory 128 (2008), 263–279.CrossRefGoogle Scholar
[19]Ward, M.Memoir on elliptic divisibility sequences. Amer. J. Math. 70 (1948), 31–74.CrossRefGoogle Scholar
[20]Zimmer, H. G.Quasifunctions on elliptic curves over local fields. J. Reine Angew. Math. 307/308 (1979), 221–246.Google Scholar
[21]Zimmer, H. G.Correction and remarks concerning quasifunctions on elliptic curves. J. Reine Angew. Math. 343 (1983), 203–211.Google Scholar