Computing rational points on rank 1 elliptic curves via 𝐿-series and canonical heights

Silverman, Joseph

doi:10.1090/S0025-5718-99-01068-6

Computing rational points
on rank 1 elliptic curves
via $L$ -series and canonical heights

Author: Joseph H. Silverman
Journal: Math. Comp. 68 (1999), 835-858
MSC (1991): Primary 11G05, 11Y50
DOI: https://doi.org/10.1090/S0025-5718-99-01068-6
MathSciNet review: 1627825
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $E/\mathbb{Q}$ be an elliptic curve of rank 1. We describe an algorithm which uses the value of and the theory of canonical heghts to efficiently search for points in $E(\mathbb{Q})$ and $E(\mathbb{Z}_{S})$ . For rank 1 elliptic curves $E/\mathbb{Q}$ of moderately large conductor (say on the order of $10^{7}$ to $10^{10}$ ) and with a generator having moderately large canonical height (say between 13 and 50), our algorithm is the first practical general purpose method for determining if the set $E(\mathbb{Z}_{S})$ contains non-torsion points.

1. C. Batut, D. Bernardi, H. Cohen, M. Olivier, PARI-GP, a computer system for number theory, Version 1.39.
2. B.J. Birch, H.P.F. Swinnerton-Dyer, Elliptic curves and modular functions, Modular Functions of One Variable IV (B.J. Birch, W. Kuyk, eds.), Lecture Notes in Math. 476, Springer-Verlag, Berlin, 1975. MR 52:5685
3. A. Bremner, On the equation $Y^{2}=X(X^{2}+p)$ , Number Theory and Applications (R.A. Mollin, ed.), Kluwer Academic Publishers, 1989, pp. 3-22. MR 92h:11047
4. A. Bremner, J.W.S. Cassels, On the equation $Y^{2}=X(X^{2}+p)$ , Math. Comp. 42 (1984), 257-264. MR 85f:11017
5. H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Math., vol. 138, Springer Verlag, Berlin, 1993. MR 94i:11105
6. J. Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press, Cambridge, 1992. MR 93m:11053
7. -, private communication, November 1995.
8. F. Diamond, On deformation rings and Hecke rings, Annals of Math. 144 (1996), 137-166. MR 97d:11172
9. N. Elkies, Heegner point computations, Algorithmic Number Theory (L.M. Adelman, M.-D. Huang, eds.), ANTS-I, Lecture Notes in Computer Science, vol. 877, 1994, pp. 122-133. MR 96f:11080
10. B. Gross and D. Zagier, Heegner points and derivatives of $L$ -series, Invent. Math. 84 (1986), 225-320. MR 87j:11057
11. V.A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, Birkhäuser, Boston, 1990, pp. 435-483. MR 92g:11109
12. D. Rohrlich, Variation of the root number in families of elliptic curves, Compositio Math. 87 (1993), 119-151. MR 94d:11045
13. K. Rubin, $p$ -adic $L$ -functions and rational points on elliptic curves with complex multiplication, Invent. Math. 107 (1992), 323-350. MR 92m:11063
14. J.H. Silverman, The Néron-Tate Height on Elliptic Curves, Ph.D. thesis, Harvard, 1981.
15. -, The Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 106, Springer-Verlag, Berlin and New York, 1986. MR 87g:11070
16. -, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Math., vol. 151, Springer-Verlag, Berlin and New York, 1994. MR 96b:11074
17. -, The difference between the Weil height and the canonical height on elliptic curves, Math. Comp. 55 (1990), 723-743. MR 91d:11063
18. R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Annals of Math. 141 (1995), 553-572. MR 96d:11072
19. D. Ulmer, A construction of local points on elliptic curves over modular curves, International Math. Research Notes 7 (1995), 349-363. MR 97b:11076
20. A. Wiles, Modular elliptic curves and Fermat's last theorem, Annals of Math. 141 (1995), 443-551. MR 96d:11071
21. D. Zagier, private communication.