Regulators of rank one quadratic twists
Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 601-624.

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.

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.

DOI : 10.5802/jtnb.643
Christophe Delaunay 1 ; Xavier-François Roblot 1

1 Université de Lyon Université Lyon 1 INSA de Lyon, F-69621 École Centrale de Lyon CNRS, UMR5208, Institut Camille Jordan 43 blvd du 11 novembre 1918 F-69622 Villeurbanne-Cedex, France
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Christophe Delaunay; Xavier-François Roblot. Regulators of rank one quadratic twists. Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 601-624. doi : 10.5802/jtnb.643. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.643/

[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 | Zbl

[Coh1] H. Cohen, A course in Computational Algebraic Number Theory. Graduate texts in Math. 138, Springer-Verlag, New-York (1993). | MR | Zbl

[Coh2] H. Cohen, Diophantine equations, p-adic Numbers and L-functions. Graduate Texts in Mathematics 239 and 240, Springer-Verlag. | MR

[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 | Zbl

[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 | Zbl

[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

[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

[De2] C. Delaunay, Moments of the Orders of Tate-Shafarevich groups. International Journal of Number Theory, 1 (2005), no. 2, 243–264. | MR | Zbl

[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. | MR | Zbl

[Elk] N. Elkies, Heegner point computations. Algorithmic number theory (Ithaca, NY, 1994), 122–133, Lecture Notes in Comput. Sci., 877, Springer, Berlin, 1994. | MR | Zbl

[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. | MR | Zbl

[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 | Zbl

[Kri] M. Krir, À propos de la conjecture de Lang sur la minoration de la hauteur de Néron-Tate pour les courbes elliptiques sur . Acta Arithmetica, C (2001), no. 1, 1–16. | MR | Zbl

[Gro-Zag] B. Gross, D. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84, (1986), 225–320. | MR | Zbl

[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 |Ш| values on quadratic twists of elliptic curves. Experiment. Math. 15 (2006), no. 3, 355–365. | MR

[Ri-Vi] G. Ricotta, T. Vidick, Hauteur Asymptotique des points de Heegner. To appear in Canad. J. Math. | MR

[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 | Zbl

[Sna] N. C. Snaith, Derivatives of random matrix characteristic polynomials with applications to elliptic curves. J. Phys. A 38 (2005), 48, 10345–10360. | MR | Zbl

[Wat] M. Watkins, Extra rank for odd parity twists, available at http://www.maths.bris.ac.uk/~mamjw/papers/papers.html

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