Computing rational points on rank 1 elliptic curves via $L$-series and canonical heights
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- by Joseph H. Silverman PDF
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Abstract:
Let $E/\mathbb {Q}$ be an elliptic curve of rank 1. We describe an algorithm which uses the value of $L’(E,1)$ 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.References
- C. Batut, D. Bernardi, H. Cohen, M. Olivier, PARI-GP, a computer system for number theory, Version 1.39.
- H. P. F. Swinnerton-Dyer and B. J. Birch, Elliptic curves and modular functions, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 2–32. MR 0384813
- A. Bremner, On the equation $Y^2=X(X^2+p)$, Number theory and applications (Banff, AB, 1988) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 265, Kluwer Acad. Publ., Dordrecht, 1989, pp. 3–22. MR 1123066
- A. Bremner and J. W. S. Cassels, On the equation $Y^{2}=X(X^{2}+p)$, Math. Comp. 42 (1984), no. 165, 257–264. MR 726003, DOI 10.1090/S0025-5718-1984-0726003-4
- Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
- J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
- —, private communication, November 1995.
- Fred Diamond, On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), no. 1, 137–166. MR 1405946, DOI 10.2307/2118586
- Noam D. Elkies, Heegner point computations, Algorithmic number theory (Ithaca, NY, 1994) Lecture Notes in Comput. Sci., vol. 877, Springer, Berlin, 1994, pp. 122–133. MR 1322717, DOI 10.1007/3-540-58691-1_{4}9
- Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809
- V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 435–483. MR 1106906
- David E. Rohrlich, Variation of the root number in families of elliptic curves, Compositio Math. 87 (1993), no. 2, 119–151. MR 1219633
- Karl Rubin, $p$-adic $L$-functions and rational points on elliptic curves with complex multiplication, Invent. Math. 107 (1992), no. 2, 323–350. MR 1144427, DOI 10.1007/BF01231893
- J.H. Silverman, The Néron-Tate Height on Elliptic Curves, Ph.D. thesis, Harvard, 1981.
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
- Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
- Joseph H. Silverman, The difference between the Weil height and the canonical height on elliptic curves, Math. Comp. 55 (1990), no. 192, 723–743. MR 1035944, DOI 10.1090/S0025-5718-1990-1035944-5
- Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
- Douglas L. Ulmer, A construction of local points on elliptic curves over modular curves, Internat. Math. Res. Notices 7 (1995), 349–363. MR 1350688, DOI 10.1155/S1073792895000262
- Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
- D. Zagier, private communication.
Additional Information
- Joseph H. Silverman
- Affiliation: Mathematics Department, Box 1917, Brown University, Providence, RI 02912 USA
- MR Author ID: 162205
- ORCID: 0000-0003-3887-3248
- Email: jhs@gauss.math.brown.edu
- Received by editor(s): May 8, 1996
- Received by editor(s) in revised form: March 3, 1997
- Additional Notes: Research partially supported by NSF DMS-9424642.
- © Copyright 1999 American Mathematical Society
- 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