staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Christophe Delaunay; Xavier-François Roblot
Regulators of rank one quadratic twists
Journal de théorie des nombres de Bordeaux, 20 no. 3 (2008), p. 601-624, doi: 10.5802/jtnb.643
Article PDF | Reviews MR 2523310 | Zbl pre05572694

Résumé - Abstract

We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of a rank one quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.

Bibliography

[An-Bu-Fr] J. A. Antoniadis, M. Bungert, G. Frey, Properties of twists of elliptic curves. J. Reine Angew. Math. 405 (1990), 1–28.  MR 1040993 |  Zbl 0709.14020
[Coh1] H. Cohen, A course in Computational Algebraic Number Theory. Graduate texts in Math. 138, Springer-Verlag, New-York (1993).  MR 1228206 |  Zbl 0786.11071
[Coh2] H. Cohen, Diophantine equations, $p$-adic Numbers and $L$-functions. Graduate Texts in Mathematics 239 and 240, Springer-Verlag.  MR 2312337
[CKRS] J. B. Conrey, J. P. Keating, M. O. Rubinstein, N. C. Snaith, On the frequency of vanishing of quadratic twists of modular $L$-functions. Number theory for the millennium, I (Urbana, IL, 2000), 301–315, A. K. Peters, Natick, MA, 2002.  MR 1956231 |  Zbl 1044.11035
[CFKRS] J. B. Conrey, D. W. Farmer J. P. Keating, M. O. Rubinstein, N. C. Snaith, Integral moments of $L$-functions. Proc. London Math. Soc. (3) 91 (2005), no. 1, 33–104.  MR 2149530 |  Zbl 1075.11058
[CRSW] J. B. Conrey, M. O. Rubinstein, N. C. Snaith, M. Watkins, Discretisation for odd quadratic twists, in Ranks of elliptic curves and random matrix theory, ed. J. B. Conrey, D. W. Farmer, F. Mezzadri and N. C. Snaith, London Mathematical Society, Lecture notes series 341, 201–214.  MR 2322346 |  Zbl pre05190713
[De1] C. Delaunay, Heuristics on class groups and on Tate-Shafarevitch groups, in Ranks of elliptic curves and random matrix theory, ed. J. B. Conrey, D. W. Farmer, F. Mezzadri and N. C. Snaith, London Mathematical Society, Lecture notes series 341, 323–340.  MR 2322335 |  Zbl pre05190722
[De2] C. Delaunay, Moments of the Orders of Tate-Shafarevich groups. International Journal of Number Theory, 1 (2005), no. 2, 243–264.  MR 2173383 |  Zbl 1082.11042
[De-Du] C. Delaunay, S. Duquesne, Numerical Investigations Related to the Derivatives of the $L$-series of Certain Elliptic Curves. Exp. Math. 12 (2003), no. 3, 311–317.
Article |  MR 2034395 |  Zbl 1083.11041
[Elk] N. Elkies, Heegner point computations. Algorithmic number theory (Ithaca, NY, 1994), 122–133, Lecture Notes in Comput. Sci., 877, Springer, Berlin, 1994.  MR 1322717 |  Zbl 0837.14044
[Hay] Y. Hayashi, The Rankin’s $L$-function and Heegner points for general discriminants. Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), no. 2, 30–32.
Article |  MR 1326793 |  Zbl 0853.11041
[K-S] J. P. Keating, N. C. Snaith, Random matrix theory and $L$-functions at $s=1/2$. Comm. Math. Phys. 214 (2000), 91–110.  MR 1794267 |  Zbl 1051.11047
[Kri] M. Krir, À propos de la conjecture de Lang sur la minoration de la hauteur de Néron-Tate pour les courbes elliptiques sur $\mathbb{Q}$. Acta Arithmetica, C (2001), no. 1, 1–16.  MR 1864622 |  Zbl 0981.11021
[Gro-Zag] B. Gross, D. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84, (1986), 225–320.  MR 833192 |  Zbl 0608.14019
[PARI] C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, PARI/GP System, available at http://pari.math.u-bordeaux.fr/
[Qua] P. Quattrini, On the distribution of analytic $\sqrt{\vert \Sha\vert }$ values on quadratic twists of elliptic curves. Experiment. Math. 15 (2006), no. 3, 355–365.
Article |  MR 2264472 |  Zbl pre05142616
[Ri-Vi] G. Ricotta, T. Vidick, Hauteur Asymptotique des points de Heegner. To appear in Canad. J. Math.  MR 2462452 |  Zbl pre05382118
[Rub] M. Rubinstein, Numerical data, available at http://www.math.uwaterloo.ca/~mrubinst/
[Sil] J. H. Silverman, The Arithmetic of Elliptic Curves. Graduate text in Math. 106, Springer-Verlag, New-York (1986).  MR 817210 |  Zbl 0585.14026
[Sna] N. C. Snaith, Derivatives of random matrix characteristic polynomials with applications to elliptic curves. J. Phys. A 38 (2005), 48, 10345–10360.  MR 2185940 |  Zbl 1086.15026
[Wat] M. Watkins, Extra rank for odd parity twists, available at http://www.maths.bris.ac.uk/~mamjw/papers/papers.html