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Abstract:

The aim of this paper is to analyze the distribution of analytic (and signed) square roots of X values on imaginary quadratic twists of elliptic curves. Given an elliptic curve E of rank zero and prime conductor N, there is a weight-(formula presented) modular form g associated with it such that the d-coefficient of g is related to the value at s = 1 of the L-series of the (−d)-quadratic twist of the elliptic curve E. Assuming the Birch and Swinnerton-Dyer conjecture, we can then calculate for a large number of integers d the order of X of the (−d)-quadratic twist of E and analyze their distribution. © A K Peters, Ltd.

Registro:

Documento: Artículo
Título:On the distribution of analytic (Formula presented) values on quadratic twists of elliptic curves
Autor:Quattrini, P.L.
Filiación:Universidad de Buenos Aires, Departamento de Matematica, FCEyN, Pab. I, Ciudad Universitaria, Buenos Aires, 1428, Argentina
Palabras clave:Elliptic curves; Modular forms; Tate-Shafarevich groups
Año:2006
Volumen:15
Número:3
Página de inicio:355
Página de fin:365
DOI: http://dx.doi.org/10.1080/10586458.2006.10128970
Título revista:Experimental Mathematics
Título revista abreviado:Exp. Math.
ISSN:10586458
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10586458_v15_n3_p355_Quattrini

Referencias:

  • Cohen, H., Lenstra, H.W., Heuristics on Class Groups of Number Fields (1984) Lecture Notes in Math, pp. 33-62. , Number Theory, Berlin: Springer-Verlag
  • Conrey, J.B., Keating, J.P., Rubinstein, M.O., Snaith, N.C., (2004) Random Matrix Theory and the Fourier Coefficients of Half-Integral Weight Forms, , arXiv: math.NT/0412083 v2
  • Cremona, J.E., Computing the Degree of the Modular Parametrization of a Modular Elliptic Curve (1995) Math. Comp., 64 (211), pp. 1235-1250
  • Delaunay, C., Heuristics on Tate-Shafarevitch Groups of Elliptic Curves Defined over Q (2001) Exper. Math, 10 (2), pp. 191-196
  • Gross, B., Heights and the Special Values of LSeries (1987) CMS Conference Proceedings, p. 7. , Providence: American Math. Soc
  • Pacetti, A., (2001) Qalgmodforms, , http://www.ma.utexas.edu/users/villegas/cnt/cnt-frames.html
  • Pizer, A., An Algorithm for Computing Modular Forms on Γ0(N) (1980) Journal of Algebra, 64, pp. 340-390
  • Rubinstein, M.O., (2002) Private Communication
  • Shimura, G., On Modular Forms of Half Integral Weight (1973) Ann. Math. Sec. Ser., 97 (3), pp. 440-481
  • Tornaria, G., (2004) Data about the Central Values of the L-Series of (Imaginary and Real) Quadratic Twists of Elliptic Curves, , http://www.ma.utexas.edu/users/tornaria/cnt

Citas:

---------- APA ----------
(2006) . On the distribution of analytic (Formula presented) values on quadratic twists of elliptic curves. Experimental Mathematics, 15(3), 355-365.
http://dx.doi.org/10.1080/10586458.2006.10128970
---------- CHICAGO ----------
Quattrini, P.L. "On the distribution of analytic (Formula presented) values on quadratic twists of elliptic curves" . Experimental Mathematics 15, no. 3 (2006) : 355-365.
http://dx.doi.org/10.1080/10586458.2006.10128970
---------- MLA ----------
Quattrini, P.L. "On the distribution of analytic (Formula presented) values on quadratic twists of elliptic curves" . Experimental Mathematics, vol. 15, no. 3, 2006, pp. 355-365.
http://dx.doi.org/10.1080/10586458.2006.10128970
---------- VANCOUVER ----------
Quattrini, P.L. On the distribution of analytic (Formula presented) values on quadratic twists of elliptic curves. Exp. Math. 2006;15(3):355-365.
http://dx.doi.org/10.1080/10586458.2006.10128970