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Efficient Computation of p-Adic Heights

Published online by Cambridge University Press:  01 February 2010

David Harvey
David Harvey
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford St, Cambridge, MA 02138, USA, [email protected]

Abstract

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We analyse and drastically improve the running time of the algorithm of Mazur, Stein and Tate for computing the canonical cyclotomic p-adic height of a point on an elliptic curve E/Q, where E has good ordinary reduction at p ≥ 5.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 11 , 2008 , pp. 40 - 59
Copyright
Copyright © London Mathematical Society 2008

References

1.Bernstein, D., ‘Fast multiplication and its applications’. 52, 58 http://cr.yp.to/lineartime/multapps-20041007.pdfGoogle Scholar
2.Brent, R. P., ‘Multiple-precision zero-finding methods and the complexity of elementary function evaluation’, Analytic computational complexity, Pittsburgh, 1975 (Academic Press, New York, 1976) 151–176.Google Scholar
3.Brent, R. P. and Kung, H. T., ‘Fast algorithms for manipulating formal power series’, J. Assoc. Comput. Mach. 25 (1978) 581–595.CrossRefGoogle Scholar
4.Harvey, D., ‘Kedlaya's algorithm in larger characteristic’, Int. Math. Res. Notices (2007), no. rnm095, rnm095–29.Google Scholar
5.Kedlaya, K. S., ‘Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology’, J. Ramanujan Math. Soc. 16 (2001) 323–338.Google Scholar
6.Lang, S., Elliptic curves: Diophantine analysis, vol. 231 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (Springer-Verlag, Berlin, 1978).CrossRefGoogle Scholar
7.Mazur, B., Stein, W. and Tate, J., ‘Computation of p-adic heights and log convergence’, Documenta Math. (Extra Volume: John H. Coates' Sixtieth Birthday) (2006) 577–614.Google Scholar
8.Mazur, B. and Tate, J., ‘The p-adic sigma function’, Duke Math. J. 62 (1991) 663–688.CrossRefGoogle Scholar
9.Silverman, J. H., The arithmetic of elliptic curves, vol. 106 of Graduate Texts in Mathematics (Springer-Verlag, New York, 1992). Corrected reprint of the 1986 original.Google Scholar
10.Stein, W. and Joyner, D., ‘Sage: System for algebra and geometry experimentation’, Communications in Computer Algebra (ACM SIGSAM Bulletin) 39 (2005) 61–64. 56, 58 http://www.sagemath.org/.Google Scholar
11.Wuthrich, C., ‘The fine Selmer group and height pairings’, Ph.D. thesis, Cambridge, (2004). 52 http://www.maths.nottingham.ac.uk/personal/cw/thesisandessays.htmlGoogle Scholar