Rational points near curves and small nonzero |x3 - y2| via lattice reduction

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 1838, Issue , 2000, Pages 33-63

  Elkies, Noam D ^a  

^a HARVARD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATION THEORY; NUMBER THEORY;

COMPUTATIONAL COSTS; INTEGER SOLUTIONS; LATTICE REDUCTION; LINEAR APPROXIMATIONS; NATURAL GENERALIZATION; PLANE CURVES; RATIONAL POINTS;

APPROXIMATION ALGORITHMS;

EID: 84946823013     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/10722028_2     Document Type: Conference Paper

References (38)

1. Bernstein, D.J.: Enumerating solutions to p(a) + q(b) = r(c) + s(d). Math, of Computation, to appear.
2. Bremner, A.: Sums of three cubes. Pages 87-91 in Number Theory (Halifax, Nova Scotia, 1994) (CMS Conf. Proc. 15), Providence: AMS, 1995.
3. 2, Norske Vid. Selsk. Torh. 38 (1965), 65-69.
4. Bombieri, E., Pila, J.: The number of integral points on arcs and ovals. Duke Math. J. 59 (1989) #2, 337-357.
5. Bruce, J.W., Wall, C.T.C.: On the classification of cubic surfaces. J. London Math. Soc. (2) 19 (1979) #2, 245-256.
6. Cohn, H.L.: New Bounds on Sphere Tackings. Ph.D. thesis, Harvard 2000.
7. Cremona, J.E.: Algorithms for modular elliptic curves. Cambridge University Press, 1992.
8. Crux Mathematicorum 8 (1982).
9. Conway, J.H., Sloane, N.J.A.: Sphere Tackings, Lattices and Groups. New York: Springer 1993.
10. Conn, W., Vaserstein, L.N.: On sums of integral cubes. Pages 285-294 in The Rademacher legacy to mathematics (University Park, 1992), Contemp. Math. 166, AMS 1994.
11. 2 = k and Hall’s conjecture, Math. Notes Acad. Sci. USSR 32 (1982), 617-618.
12. 2(t), Norske Vid. Selsk. Torh. 38 (1965), 86-87.
13. 4, Math, of Computational (Oct.88) #184, 825-835.
14. Elkies, N.D.: ABC implies Mordell, International Math. Research Notices 1991 #7, 99-109.
15. Elkies, N.D.: Heegner point computations. Pages 122-133 in Algorithmic Number Theory (Proceedings of ANTS-I; L.M. Adleman, M.-D. Huang, eds.; Berlin: Springer, 1994; Lecture Notes in Computer Science 877).
16. Elkies, N.D.: Elliptic and modular curves over finite fields and related computational issues. Pages 21-76 in Computational Terspectives on Number Theory: Proceedings of a Conference in Honor of A.O.L. Atkin (D.A. Buell and J.T. Teitelbaum, eds.; AMS/International Press, 1998).
17. Eikies, N.D.: Shimura curve computations. Pages 1-47 in Algorithmic Number Theory (Proceedings of ANTS-III; J. P. Buhler, ed.; Berlin: Springer, 1998; Lecture Notes in Computer Science 1423).
18. Fricke, R.: Ueber eine einfache Gruppe von 504 Operationen, Math. Ann. 52 (1899), 321-339.
19. Fulton, W., Harris, J.: Representation Theory: A Tirst Course. New York: Springer, 1991 (GTM 129).
20. Gebel, J., Pethö, A., and Zimmer, H.G.: On Mordell’s equation, Compositio Math. 110 (1998), 335-367.
21. Guy, R.K.: Unsolved Problems in Number Theory. New York: Springer, 1981.
22. 2 = k. Pages 173-198 in Computers in Number Theory (A. Atkin, B. Birch, eds.; Academic Press, 1971).
23. Heath-Brown, D.R.: The density of zeros of forms for which weak approximation fails. Math, of Computation 59 (1992) #200, 613-623.
24. Heath-Brown, D.R.: The density of rational points on projective hypersurfaces. Preprint, 2000.
25. 3 = k on a vector computer. Math, of Computation 61 (1993) #203, 235-244.
26. Keller, W., Kulesz, L.: Courbes algébriques de genre 2 et 3 possédant de nombreux points rationnels. G. R. Acad. Sei. Paris, Sér. I Math. 321 (1995) #11, 1469-1472.
27. 3 = n. Math, of Computation 66 (1997) #218, 841-851.
28. Lang, S.: Old and new conjectured diophantine inequalities, Bull. Amer. Math. Society #23 (1990), 37-75.
29. Macbeath, A.M.: On a curve of genus 7, Proc. London Math. Soc. 15 (1965), 527-542.
30. Mahler, K.: Lectures on Transcendental Numbers. Berlin: Springer, 1976; Lecture Notes in Math. 546.
31. Mason, R.C.: Diophantine Equations over Tunction Tields, London Math. Soc. Lecture Notes Series #96, Cambridge Univ. Press, 1984. See also pages 149157 in Springer LNM 1068 (1984) [=proceedings of Journées Arithmétiques 1983, Noordwijkerhout].
32. Oesterlé, J.: Nouvelles approches du "théorème" de Fermat, Sém. Bourbaki 2/88, exposé #694.
33. Pila, J.: Geometric postulation of a smooth function and the number of rational points, Duke Math. J. 63 (1991), 449-463.
34. Payne, G., Vaserstein, L.N.: Sums of three cubes. Pages 443-454 in The Arithmetic of Tunction Tields, de Gruyter, 1992.
35. Stahlke, C.: Algebraic curves over ℚ with many rational points and minimal automorphism group. Lnternational Math. Research Notices 1997 #1, 1-4.
36. Takeuchi, K.: Commensurability classes of arithmetic triangle groups, J. Tac. Set. Univ. Tokyo 24 (1977), 201-212.
37. Weil, A.: Abstract versus classical algebraic geometry. Pages 550-558 of Proceedings of the Lnternational Congress of Mathematicians, 1954, Amsterdam, Vol. LLL.
38. Wildanger, K.: Über das Lösen von Einheiten- und Lndexformgleichungen in algebraischen Zahlkörpern mit einer Anwendung auf die Bestimmung aller ganzen Punkte einer Mordellschen Kurve. Ph.D. Thesis, TU Berlin, Berlin, 1997.