58
Views
14
CrossRef citations to date
0
Altmetric
Original Article

Some Heuristics about Elliptic Curves

Pages 105-125 | Published online: 30 Jan 2011

  • B. J. Birch. "How the Number of Points of an Elliptic Curve over a Fixed Prime Field Varies." J. London Math. Soc. 43 (1968), 57–60.
  • B. J. Birch and H. P. F. Swinnerton-Dyer. "Notes on Elliptic Curves, I." J. Reine Angew. Math. 212 (1963), 7–25.
  • B. J. Birch and H. P. F. Swinnerton-Dyer. "Notes on Elliptic Curves, II." J. Reine Angew. Math. 218 (1965), 79–108.
  • C. Breuil, B. Conrad, F. Diamond, and R. Taylor. "On the Modularity of Elliptic Curves over Q: Wild 3-adic Exercises." J. Amer. Math. Soc. 14:4 (2001), 843-939.
  • N. G. de Bruijn. "On the Number of Integers ≤ x Whose Prime Factors Divide n." Illinois J. Math. 6 (1962), 137–141.
  • A. Brumer. "The Average Rank of Elliptic Curves, I." Invent. Math. 109:3 (1992), 445–472.
  • A. Brumer and O. McGuinness. "The behavior of the Mordell–Weil Group of Elliptic Curves." Bull. Amer. Math. Soc. (N.S.) 23:2 (1990), 375-382.
  • J. Buhler, C. Schoen, and J. Top. "Cycles, L-Functions and Triple Products of Elliptic Curves." J. Reine Angew. Math. 492 (1997), 93–133.
  • S. Burris and K. Yeats. "Admissible Dirichlet Series." Preprint available online (arxiv.org/ math/0507487), 2005.
  • H. Cohen. A Course in Computational Algebraic Number Theory, Grad. Texts in Math., 138. New York: Springer-Verlag, 1993.
  • I. Connell. The Elliptic Curve Handbook, Lecture Notes from a course taught at McGill University. Available online (www.math.mcgill.ca/connell/public/ECH1), 1991.
  • B. Conrad, F. Diamond, and R. Taylor. "Modularity of Certain Potentially Barsotti–Tate Galois Representations." J. Amer. Math. Soc. 12:2 (1999), 521-567.
  • J. B. Conrey, J. P. Keating, M. O. Rubinstein, and N. C. Snaith. "On the Frequency of Vanishing of Quadratic Twists of Modular L-Functions." In Number Theory for the Millennium, I (Urbana, IL, 2000), pp. 301-315. Natick, MA: A K Peters, 2002.
  • J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith. "Integral Moments of L-Functions." Proc. London Math. Soc. (3) 91:1 (2005), 33–104.
  • J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith. "Random Matrix Theory and the Fourier Coefficients of Half-Integral Weight Forms." Experimental Math. 15:1 (2006), 67–82.
  • J. B. Conrey, A. Pokharel, M. O. Rubinstein, and M. Watkins. "Secondary Terms in the Number of Vanishings of Quadratic Twists of Elliptic Curve LFunctions." In Ranks of Elliptic Curves and Random Matrix Theory, edited by J. B. Conrey, D. W. Farmer, F. Mezzadri, and N. C. Snaith, pp. 215–232, London Mathematical Society Lecture Note Series 341. Cambridge: Cambridge University Press, 2007.
  • J. E. Cremona. mwrank (software). Available online (www.warwick.ac.uk/~masgaj/ftp/progs), 2005.
  • J. E. Cremona. Elliptic Curve Data. Available online (www.warwick.ac.uk/~masgaj/ftp/data/), 2006.
  • C. Delaunay and M. Watkins. "The Powers of Logarithm for Quadratic Twists." in Ranks of Elliptic Curves and Random Matrix Theory, edited by J. B. Conrey, D. W. Farmer, F. Mezzadri, and N. C. Snaith, pp. 189–193, London Mathematical Society Lecture Note Series 341. Cambridge: Cambridge University Press, 2007.
  • F. Diamond. "On Deformation Rings and Hecke Rings." Ann. of Math. (2) 144:1 (1996), 137–166.
  • T. Dokchiter and V. Dokchitser "Root Numbers of Elliptic Curves in Residue Characteristic 2." Preprint available online (arxiv.org/math/ 0612054), 2006.
  • W. Duke. "Elliptic curves with No Exceptional Primes." C. R. Acad. Sci. Paris Sér. I Math. 325:8 (1997) 813–818.
  • W. Duke and E. Kowalski. "A Problem of Linnik for Elliptic Curves and Mean-Value Estimates for Automorphic Representations." Invent. Math. 139:1 (2000) 1–39.
  • N. Dummigan, P. Martin, and M. Watkins. "Euler Factors and Local Root Numbers for Symmetric Powers of Elliptic Curves." Preprint, 2006.
  • T. A. Fisher. "Finding Rational Points on Elliptic Curves Using 6-Descent and 12-Descent." Preprint available online (www.dpmms.cam.ac.uk/~taf1000/papers/sixandtwelve.html), 2007.
  • É. Fouvry, M. Nair, and G. Tenenbaum. "L'ensemble exceptionnel dans la conjecture de Szpiro." Bull. Soc. Math. France 120:4 (1992), 485–506.
  • G. Frey, editor. On Artin's Conjecture for Odd 2- Dimensional Representations, Lecture Notes in Mathematics, 1585. Berlin: Springer-Verlag, 1994.
  • D. Goldfeld. "Conjectures on Elliptic Curves over Quadratic Fields." In Number Theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), edited by M. B. Nathanson, pp. 108–118, Lect. Notes in Math. 751. Berlin: Springer- Verlag, 1979.
  • R. Greenberg. "On the Birch and Swinnerton- Dyer Conjecture." Invent. Math. 72:2 (1983), 241–265.
  • H. A. Helfgott. "On the Behaviour of Root Numbers in Families of Elliptic Curves." Preprint available online (arxiv.org/math/0408141), 2004.
  • M. Hindry. "Why Is It Difficult to Compute the Mordell–Weil Group?" Preprint available online (www. math.jussieu.fr/~hindry/MW-size.pdf), 2005.
  • N. M. Katz. Moments, Monodromy, and Perversity: A Diophantine Perspective, Annals of Mathematics Studies, 159. Princeton: Princeton University Press, 2005.
  • N. M. Katz and P. Sarnak. Random Matrices, Frobenius Eigenvalues, and Monodromy, American Mathematical Society Colloquium Publications, 45. Providence: American Mathematical Society, 1999.
  • J. P. Keating and N. C. Snaith. "Random Matrix Theory and ζ(1/2 + it)" and "Random Matrix Theory and L-Functions at s = 1/2." Comm. Math. Phys. 214:1 (2000), 57–89 and 91–110.
  • S. Kobayashi. "The Local Root Number of Elliptic Curves with Wild Ramification." Math. Ann. 323:3 (2002), 609–623.
  • S. Lang. "Conjectured Diophantine Estimates on Elliptic Curves." In Arithmetic and Geometry, vol. I: Arithmetic, Papers Dedicated to I. R. Shafarevich on the Occasion of his Sixtieth Birthday, edited by M. Artin and J. Tate, pp. 155–171, Progress in Mathematics 35. Boston: Birkhäuser, 1983.
  • C. Liu and L. Xu. "The Vanishing Order of Certain Hecke L-Functions of Imaginary Quadratic Fields." J. Number Theory 108:1 (2004), 76–89.
  • V. A. Marčenko and L. A. Pastur. "Distribution of Eigenvalues in Certain Sets of Random Matrices." Math. USSR-Sb. 1 (1967), 457–483. (Russian original in Mat. Sb. (N.S.) 72:114 (1967), 507–536.)
  • P. Martin and M. Watkins. "Symmetric Powers of Elliptic Curve L-Functions." In Algorithmic Number Theory, Proceedings of the 7th International Symposium, ANTS-VII, Berlin, Germany, July 2006, edited by F. Hess, S. Pauli, and M. Pohst, pp. 377–392, Springer Lecture Notes in Computer Science 4076. Berlin: Springer, 2006.
  • M. L. Mehta. Random Matrices, third edition, Pure and Applied Mathematics (Amsterdam) 142. Amsterdam: Elsevier/Academic Press, 2004.
  • P. Michel. "Rang moyen de familles de courbes elliptiques et lois de Sato–Tate." Monatsh. Math. 120:2 (1995), 127–136.
  • S. J. Miller. "One- and Two-Level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying Group Symmetries." Compos. Math. 140:4 (2004), 952–992.
  • A. Pacetti and G. Tornaría. "Computing Central Calues of Twisted L-Series: The Case of Composite Levels." To appear in Exp. Math., 2008.
  • D. Ramakrishnan and F. Shahidi. "Siegel Modular Forms of Genus 2 Attached to Elliptic Curves." Math. Res. Lett. 14:2 (2007), 315–332.
  • M. Rapoport, N. Schappacher, and P. Schneider, editors. Beilinson's Conjectures on Special Values of L-Functions, Perspectives in Mathematics, 4. Boston: Academic Press, 1988.
  • F. Rodriguez Villegas and D. Zagier. "Square Roots of Central Values of Hecke L-Series." In Advances in Number Theory, Proceedings of the Third Conference of the Canadian Number Theory Association held at Queen's University, Kingston, Ontario, August 18–24, 1991, edited by F. Q. Gouvêa and N. Yui, pp. 81–99, Oxford Sci. Publ. New York: Oxford Univ. Press, 1993.
  • D. E. Rohrlich. "The Nonvanishing of Certain Hecke L-Functions at the Center of the Critical Strip." Duke Math. J. 47:1 (1980), 223–232.
  • D. E. Rohrlich and H. L. Montgomery. "On the L-Functions of Canonical Hecke Characters of Imaginary Quadratic Fields, I, II." Duke Math. J. 47:3 (1980), 547–557, and Duke Math. J. 49:4 (1982), 937–942.
  • H. Rosson and G. Tornaría. "Central Values of Quadratic Twists for a Modular Form of Weight 4." In Ranks of Elliptic Curves and Random Matrix Theory, edited by J. B. Conrey, D. W. Farmer, F. Mezzadri, and N. C. Snaith, pp. 315–321, London Mathematical Society Lecture Note Series 341. Cambridge: Cambridge University Press, 2007.
  • W. Schwarz. "Einige Anwendungen Tauberscher Sätze in der Zahlentheorie." J. Reine Angew. Math. 219 (1965), 157–179.
  • C. L. Siegel. "A Mean Value Theorem in the Geometry of Numbers." Ann. of Math. (2) 46 (1945), 340–347.
  • J. H. Silverman. The Arithmetic of Elliptic Curves. New York: Springer-Verlag, 1992.
  • S. K. M. Stamminger. "Explicit 8-Descent on Elliptic Curves." PhD thesis, International University Bremen.
  • W. A. Stein and M. Watkins. "A Database of Elliptic Curves—First Report." In Algorithmic Number Theory, ANTS-V (Sydney 2002), edited by C. Fieker and D. R. Kohel, pp. 267–275, Springer Lecture Notes in Computer Science, 2369. New York: Springer, 2002.
  • J. T. Tate. "Algebraic Cycles and Poles of Zeta Functions." In Arithmetical Algebraic Geometry, Proceedings of a Conference at Purdue Univ. 1963, edited by O. F. G. Schilling, pp. 93–110. New York: Harper & Row, 1965.
  • R. Taylor and Wiles. "Ring-Theoretic Properties of Certain Hecke Algebras." Ann. of Math. (2) 141:3 (1995), 553–572.
  • G. Tenenbaum. "La méthode du col en théorie analytique des nombres." In Séminaire de Théorie des Nombres, Paris 1986–87, edited by C. Goldstein, pp. 411–441, Progr. Math. 75. Boston: Birkhäuser Boston, 1988.
  • J.-L. Waldspurger. "Sur les coefficients de Fourier des formes modulaires de poids demi-entier." J. Math. Pures Appl. (9) 60:4 (1981), 375–484.
  • D. Whitehouse. "Root Numbers of Elliptic Curves over 2-adic Fields." Preprint, 2004.
  • E. Wigner. "Characteristic Vectors of Bordered Matrices with Infinite Dimensions." Ann. of Math. (2) 62 (1955), 546–564.
  • A. Wiles. "Modular Elliptic Curves and Fermat's Last Theorem." Ann. of Math. (2) 141:3 (1995), 443–551.
  • J. Wishart. "The Generalized Product Moment Distribution in Samples from a Normal Multivariate Population." Biometrika 20A (1928), 32–52.
  • M. P. Young. "Low-Lying Zeros of Families of Elliptic Curves." J. Amer. Math. Soc. 19:1 (2006), 205–250.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.