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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 | Analyses MR 2523310 | Zbl pre05572694

Résumé - Abstract

Nous étudions les régulateurs des courbes elliptiques de rang 1 appartenant à des familles de tordues quadratiques d’une courbe fixée. En particulier, nous formulons des conjectures sur la taille moyenne de ces régulateurs. Nous décrivons également un algorithme performant pour calculer explicitement les invariants des tordues quadratiques de rang 1 d’une courbe elliptique (régulateur, ordre du groupe de Tate-Shafarevich, etc.) et nous comparons les données numériques obtenues avec les prédictions.

Bibliographie

[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