Explicit 4-descents on an elliptic curve

• Autorzy: J. R. Merriman , S. Siksek , N. P. Smart
• Języki publikacji: EN

Słowa kluczowe

EN

elliptic curves; Computational Number Theory

Kategorie tematyczne

• 11Y16: Algorithms; complexity
• 11G05: Elliptic curves over global fields

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1996
• Tom: 77
• Numer: 4
• Strony: 385-404

Daty

wydano

1996

otrzymano

1996-03-05

Twórcy

autor

J. R. Merriman

• Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, Kent, England

autor

S. Siksek

• Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, Kent, England

autor

N. P. Smart

• Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, Kent, England

Bibliografia

• [1] B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7-25.
• [2] B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79-108.
• [3] A. Bremner, On the equation y² = x(x²+p), in: Number Theory and Applications, R. A. Mollin (ed.), Kluwer, Dordrecht, 1989, 3-23.
• [4] A. Bremner and J. W. S. Cassels, On the equation y² = x(x²+p), Math. Comp. 42 (1984), 257-264.
• [5] J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193-291.
• [6] J. W. S. Cassels, The Mordell-Weil group of curves of genus 2, in: Arithmetic and Geometry Papers Dedicated to I. R. Shafarevich on the Occasion of his Sixtieth Birthday, Vol. 1, Birkhäuser, 1983, 29-60.
• [7] J. W. S. Cassels, Local Fields, London Math. Soc. Student Texts, Cambridge University Press, 1986.
• [8] J. W. S. Cassels, Lectures on Elliptic Curves, London Math. Soc. Student Texts, Cambridge University Press, 1991.
• [9] H. Cohen, A Course in Computational Algebraic Number Theory, Springer, Berlin, 1993.
• [10] I. Connell, Addendum to a paper of Harada and Lang, J. Algebra 145 (1992), 463-467.
• [11] J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press, 1992.
• [12] J. Gebel, A. Pethő and H. G. Zimmer, Computing integral points on elliptic curves, Acta. Arith. 68 (1994), 171-192.
• [13] J. Gebel and H. G. Zimmer, Computing the Mordell-Weil group of an elliptic curve over ℚ, in: Elliptic Curves and Related Topics, H. Kisilevsky and M. Ram Murty (eds.), CRM Proc. Lecture Notes 4, Amer. Math. Soc., 1994.
• [14] M. J. Greenberg, Lectures on Forms in Many Variables, W. A. Benjamin, 1969.
• [15] W. H. Greub, Linear Algebra, Springer, 1967.
• [16] M. J. Razar, A relation between the two component of the Tate-Šafarevič group and L(1) for certain elliptic curves, Amer. J. Math. 96 (1974), 127-144.
• [17] S. Siksek, Descents on Curves of Genus 1, PhD thesis, Exeter University, 1995.
• [18] S. Siksek, Infinite descent on elliptic curves, Rocky Mountain J. Math. 25 (1995), 1501-1538.
• [19] S. Siksek and N. P. Smart, On the complexity of computing the 2-Selmer group of an elliptic curve, preprint, 1995.
• [20] J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, 1986.
• [21] N. P. Smart, S-integral points on elliptic curves, Proc. Cambridge Philos. Soc. 116 (1994), 391-399.
• [22] N. P. Smart and N. M. Stephens, Integral points on elliptic curves over number fields, Proc. Cambridge Philos. Soc., to appear, 1996.
• [23] R. J. Stroeker and N. Tzanakis, Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms, Acta. Arith. 67 (1994), 177-196.
• [24] H. P. F. Swinnerton-Dyer, Rational zeros of two quadratic forms, Acta. Arith. 9 (1964), 261-270.
• [25] J. A. Todd, Projective and Analytical Geometry, Pitman, 1947.
• [26] A. Weil, Number Theory. An Approach Through History, Birkhäuser, 1984.