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THE NUMBER OF SOLUTIONS TO MORDELL’S EQUATION IN CONSTRAINED RANGES

Part of: Additive number theory; partitions Exponential sums and character sums

Published online by Cambridge University Press:  05 December 2014

Matthew P. Young
Matthew P. Young*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, U.S.A. email myoung@math.tamu.edu
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Abstract

We estimate the number of solutions to $|y^{2}-x^{3}|\leqslant X$ with $N\leqslant y\leqslant 2N$, in terms of both $N$ and $X$.

MSC classification

Primary: 11L07: Estimates on exponential sums 11P21: Lattice points in specified regions
Type
Research Article
Information
Mathematika , Volume 61 , Issue 3 , September 2015 , pp. 708 - 718
Copyright
Copyright © University College London 2014 

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References

Baier, S. and Browning, T., Inhomogeneous cubic congruences and rational points on del Pezzo surfaces. J. Reine Angew. Math. 680 2013, 69–151.Google Scholar
Beukers, F. and Stewart, C. L., Neighboring powers. J. Number Theory 130(3) 2010, 660–679.CrossRefGoogle Scholar
Birch, B. J., Chowla, S., Hall, M. and Schinzel, A., On the difference x 3 − y 2. Norske Vid. Selsk. Forh. (Trondheim) 38 1965, 65–69.Google Scholar
Blomer, V., Khan, R. and Young, M., Distribution of Mass of Hecke eigenforms. Duke Math. J. 162(14) 2013, 2609–2644.CrossRefGoogle Scholar
Bombieri, E. and Pila, J., The number of integral points on arcs and ovals. Duke Math. J. 59(2) 1989, 337–357.CrossRefGoogle Scholar
Brumer, A. and McGuinness, O., The behavior of the Mordell–Weil group of elliptic curves. Bull. Amer. Math. Soc. (N.S.) 23(2) 1990, 375–382.CrossRefGoogle Scholar
Brumer, A. and Silverman, J., The number of elliptic curves over Q with conductor N. Manuscripta Math. 91(1) 1996, 95–102.CrossRefGoogle Scholar
Davenport, H., On f 3(t) − g 2(t). Norske Vid. Selsk. Forh. (Trondheim) 38 1965, 86–87.Google Scholar
Dujella, A., On Hall’s conjecture. Acta Arith. 147(4) 2011, 397–402.CrossRefGoogle Scholar
Duke, W. and Kowalski, E., A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations. With an appendix by Dinakar Ramakrishnan. Invent. Math. 139(1) 2000, 1–39.CrossRefGoogle Scholar
Elkies, N., Rational points near curves and small non-zero |x 3 − y 2| via lattice reduction. In Algorithmic Number Theory (Leiden, 2000) (Lecture Notes in Computer Science 1838), Springer (Berlin, 2000), 33–63.CrossRefGoogle Scholar
Ellenberg, J. and Venkatesh, A., Reflection principles and bounds for class group torsion. Int. Math. Res. Not. IMRN 2007 2007, doi:10.1093/imrn/rnm002.CrossRefGoogle Scholar
Fouvry, É., Sur le comportement en moyenne du rang des courbes y 2 = x 3 + k. In Séminaire de Théorie des Nombres, Paris, 1990–91 (Progress in Mathematics 108), Birkhäuser (Boston, MA, 1993), 61–84.CrossRefGoogle Scholar
Fouvry, É, Nair, M. and Tenenbaum, G., L’ensemble exceptionnel dans la conjecture de Szpiro. Bull. Soc. Math. France 120(4) 1992, 485–506.CrossRefGoogle Scholar
Hall, M., The Diophantine equation x 3 − y 2 = k. In Computers in Number Theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969), Academic Press (London, 1971), 173–198.Google Scholar
Helfgott, H. and Venkatesh, A., Integral points on elliptic curves and 3-torsion in class groups. J. Amer. Math. Soc. 19(3) 2006, 527–550.CrossRefGoogle Scholar
Huxley, M., The integer points in a plane curve. Funct. Approx. Comment. Math. 37(1) 2007, 213–231.CrossRefGoogle Scholar
Huxley, M. and Trifonov, O., The square-full numbers in an interval. Math. Proc. Cambridge Philos. Soc. 119(2) 1996, 201–208.CrossRefGoogle Scholar
Lang, S., Old and new conjectured Diophantine inequalities. Bull. Amer. Math. Soc. (N.S.) 23(1) 1990, 37–75.CrossRefGoogle Scholar
Pierce, L., The 3-part of class numbers of quadratic fields. J. London Math. Soc. (2) 71(3) 2005, 579–598.CrossRefGoogle Scholar
Swinnerton-Dyer, H. P. F., The number of lattice points on a convex curve. J. Number Theory 6 1974, 128–135.CrossRefGoogle Scholar
Trifonov, O., The integer points close to a smooth curve. Serdica Math. J. 24(3-4) 1998, 319–338.Google Scholar
Watkins, M., Some heuristics about elliptic curves. Experiment. Math. 17(1) 2008, 105–125.CrossRefGoogle Scholar
Zannier, U., On Davenport’s bound for the degree of f 3 − g 2 and Riemann’s existence theorem. Acta Arith. 71(2) 1995, 107–137.CrossRefGoogle Scholar