On elliptic curves and random matrix theory
Journal de Théorie des Nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 829-845.

Rubinstein a accumulé une masse de données concernant les tordues quadratiques paires d’une courbe elliptique fixée, et comparé les résultats aux prédictions venues du modèle des matrices aléatoires. Nous utilisons la méthode des points de Heegner pour obtenir des données comparables (en nombre plus faible) pour les tordues impaires. Nous constatons de nouveau qu’au moins une des principales prédictions de la théorie des matrices aléatoires est confortée par les données.

Rubinstein has produced a substantial amount of data about the even parity quadratic twists of various elliptic curves, and compared the results to predictions from random matrix theory. We use the method of Heegner points to obtain a comparable (yet smaller) amount of data for the case of odd parity. We again see that at least one of the principal predictions of random matrix theory is well-evidenced by the data.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.653
@article{JTNB_2008__20_3_829_0,
author = {Mark Watkins},
title = {On elliptic curves and random matrix theory},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {829--845},
publisher = {Universit\'e Bordeaux 1},
volume = {20},
number = {3},
year = {2008},
doi = {10.5802/jtnb.653},
zbl = {pre05572704},
mrnumber = {2523320},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.653/}
}
Mark Watkins. On elliptic curves and random matrix theory. Journal de Théorie des Nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 829-845. doi : 10.5802/jtnb.653. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.653/

[1] B. J. Birch, N. M. Stephens, Computation of Heegner points. In Modular forms (Durham, 1983). Papers from the symposium held at the University of Durham, Durham, June 30 to July 10, 1983. Edited by R. A. Rankin. Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York (1984), 13–41. | MR 803360 | Zbl 0559.14010

[2] B. J. Birch, H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I. II. J. reine angew. Math. 212 (1963), 7–25, 218 (1965), 79–108. Available online from the Göttinger Digitalisierungszentrum via and | MR 146143 | Zbl 0147.02506

[3] W. Bosma, C. Playoust, J. Cannon, The Magma algebra system. I. The user language. In Computational algebra and number theory. Proceedings of the 1st MAGMA Conference held at Queen Mary and Westfield College, London, August 23–27, 1993. Edited by J. Cannon and D. Holt, Elsevier Science B.V., Amsterdam (1997), 235–265. Cross-referenced as J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Possibly available from sciencedirect.com through | Article | MR 1484478 | Zbl 0898.68039

[4] C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over $\mathbf{Q}$: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. Online from ams.org through | Article | MR 1839918 | Zbl 0982.11033

[5] H. Cohen, Number Theory. (Part I: Tools, and Part II: Diophantine Equations). Graduate Texts in Mathematics 239, Springer, 2007. | MR 2312337 | Zbl 1119.11001

[6] B. Conrad, Gross–Zagier revisited. With an appendix by W. R. Mann. In Heegner points and Rankin $L$-series. Papers from the Workshop on Special Values of Rankin $L$-Series held in Berkeley, CA, December 2001. Edited by H. Darmon and S.-W. Zhang. Mathematical Sciences Research Institute Publ. 49, Cambridge University Press, Cambridge (2004), 67–163. Available online from | MR 2083211 | Zbl 1072.11040

[7] J. B. Conrey, J. P. Keating, M. O. Rubinstein, 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). Papers from the conference held at the University of Illinois at Urbana–Champaign, Urbana, IL, May 21–26, 2000. Edited by M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand and W. Philipp. Published by A K Peters, Ltd., Natick, MA (2002), 301–315. Preprint at | MR 1956231 | Zbl 1044.11035

[8] J. B. Conrey, J. P. Keating, M. O. Rubinstein, N. C. Snaith, Random Matrix Theory and the Fourier Coefficients of Half-Integral Weight Forms, Exper. Math. 15 (2006), no. 1, 67–82. For data see [37]. Preprint available online from and paper at | MR 2229387 | Zbl 1144.11035

[9] J. B. Conrey, A. Pokharel, M. O. Rubinstein, and M. Watkins, Secondary terms in the number of vanishings of quadratic twists of elliptic curve $L$-functions. In Ranks of Elliptic Curves and Random Matrix Theory, edited by J. B. Conrey, D. W. Farmer, F. Mezzadri, and N. C. Snaith, London Mathematical Society Lecture Note Series 341, Cambridge University Press (2007), 215–232. Preprint at | MR 2322347 | Zbl pre05190714

[10] 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, edited by J. B. Conrey, D. W. Farmer, F. Mezzadri, and N. C. Snaith, London Mathematical Society Lecture Note Series 341, Cambridge University Press (2007), 201–214. Preprint online at | MR 2322346 | Zbl pre05190713

[11] C. Delaunay, Moments of the orders of Tate-Shafarevich groups. Int. J. Number Theory 1 (2005), no. 2, 243–264. Possibly available online from www.worldscinet.com via | Article | MR 2173383 | Zbl 1082.11042

[12] C. Delaunay, Note on the frequency of vanishing of $L$-functions of elliptic curves in a family of 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, London Mathematical Society Lecture Note Series 341, Cambridge University Press (2007), 195–200. | MR 2322345 | Zbl pre05190712

[13] C. Delaunay, S. Duquesne, Numerical investigations related to the derivatives of the $L$-series of certain elliptic curves. Exper. Math. 12 (2003), no. 3, 311–317. Online at (sic) | MR 2034395 | Zbl 1083.11041

[14] C. Delaunay, X.-F. Roblot, Regulators of rank one quadratic twists. Preprint (2007), online at

[15] C. Delaunay, 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, London Mathematical Society Lecture Note Series 341, Cambridge University Press (2007), 189–193. | MR 2322344 | Zbl pre05190711

[16] N. D. Elkies, Heegner point computations. In Algorithmic Number Theory. Proceedings of the First International Symposium (ANTS-I) held at Cornell University, Ithaca, New York, May 6–9, 1994, edited by L. M. Adleman and M.-D. Huang, Lecture Notes in Computer Science 877, Springer-Verlag, Berlin (1994), 122–133. • N. D. Elkies, Curves $D{y}^{2}={x}^{3}-x$ of odd analytic rank. In Algorithmic number theory. Proceedings of the 5th International Symposium (ANTS-V) held at the University of Sydney, Sydney, July 7–12, 2002, edited by C. Fieker and D. R. Kohel, Lecture Notes in Computer Science 2369, Springer-Verlag, Berlin (2002), 244–251. Preprint available online from and possibly the paper from springerlink.com via ◇ The data are available online from and | Article | MR 1322717 | Zbl 0837.14044

[17] T. Fisher, Some examples of 5 and 7 descent for elliptic curves over Q. J. Eur. Math. Soc. (JEMS) 3 (2001), no. 2, 169–201. Possibly available from springerlink.com via | Article | MR 1831874 | Zbl 1007.11031

[18] C. F. Gauss, Theoria motus corporum coelestium in sectionibus conicis solem ambientium. (Latin) [Theory of motion of celestial bodies revolving about the sun in conic sections]. Published in 1809, with priority for least squares claimed from 1795 in §186. A modern English translation is Theory of the Motions of the Heavenly Bodies Moving about the Sun in Conic Sections, Dover, 2004. Original available online from the the Göttinger Digitalisierungszentrum via | Zbl 0114.24306

[19] D. Goldfeld, Conjectures on elliptic curves over quadratic fields. In Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), M. B. Nathanson, ed., Lecture Notes in Math. 751, Springer-Verlag, Berlin (1979), 108–118. | MR 564926 | Zbl 0417.14031

[20] B. H. Gross, D. B. Zagier, Heegner points and derivatives of $L$-series. Invent. Math. 84 (1986), no. 2, 225–320. Available online from the Göttinger Digitalisierungszentrum (digital library) via or possibly from springerlink.com through | Article | MR 833192 | Zbl 0608.14019

[21] J. Hadamard, Sur la distribution des zéros de la fonction $\zeta \left(s\right)$ et ses conséquences arithmétiques. (French) [On the distribution of zeros of the function $\zeta \left(s\right)$ and its arithmetic consequences]. Bull. Soc. Math. France 24 (1896), 199–220. Available online from | Numdam | MR 1504264

[22] Y. Hayashi, Die Rankinsche $L$-Funktion und Heegner-Punkte für allgemeine Diskriminanten. (German) [The Rankin $L$-function and Heegner points for general discriminants]. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 1993. Bonner Mathematische Schriften 259, Universität Bonn, Mathematisches Institut, Bonn, 1994, viii+157pp. • 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. Available online from projecteuclid.org at | MR 1293963 | Zbl 0836.11018

[23] K. Heegner, Diophantische Analysis und Modulfunktionen. (German) [Diophantine analysis and modular functions]. Math. Z. 56 (1952), 227–253. Available online from the Göttinger Digitalisierungszentrum via or possibly from springerlink.com via | Article | MR 53135 | Zbl 0049.16202

[24] N. Jochnowitz, A $p$-adic conjecture about derivatives of $L$-series attached to modular forms. In $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture. Papers from the workshop held at Boston University, Boston, Massachusetts, August 12–16, 1991, edited by B. Mazur and G. Stevens, Contemporary Mathematics 165, American Mathematical Society, Providence, RI (1994), 239–263. | MR 1279612 | Zbl 0869.11040

[25] A. M. Legendre, Nouvelles méthodes pour la détermination des orbites des comètes; avec un appendice: Sur la Méthode des moindres quarrés. (French) [New methods for the determination of the orbits of comets; with an appendix: On the method of least squares]. Published in 1805. Available online from • M. Legendre, Sur la méthode moindres quarrés, et sur l’attraction des ellipsoïdes homogènes. (French) [On the method of least squares, and the attraction of homogeneous ellipsoids]. Republication of relevant parts of above in Mem. Acad., 1811. Available online from

[26] T. Lundy, J. Van Buskirk, A new matrix approach to real FFTs and convolutions of length ${2}^{k}$. Computing 80 (2007), no. 1, 23–45. Possibly available from springerlink.com via | Article | MR 2308835 | Zbl 1138.65112

[27] Z. Mao, F. Rodriguez-Villegas, G. Tornaría, Computation of the central value of quadratic twists of modular $L$-functions. In Ranks of Elliptic Curves and Random Matrix Theory, edited by J. B. Conrey, D. W. Farmer, F. Mezzadri, and N. C. Snaith, London Mathematical Society Lecture Note Series 341, Cambridge University Press (2007), 273–288. Preprint available from | MR 2322352 | Zbl pre05190719

[28] P. Martin, M. Watkins, Symmetric powers of elliptic curve $L$-functions. In Algorithmic Number Theory. Proceedings of the 7th International Symposium (ANTS-VII) held at the Technische Universität Berlin, Berlin, July 23–28, 2006, edited by F. Hess, S. Pauli, and M. Pohst, Lecture Notes in Computer Science 4076, Springer, Berlin (2006), 377–392. Preprint available at and possibly the paper from springerlink.com via | Article | MR 2282937 | Zbl 1143.11332

[29] J.-F. Mestre, J. Oesterlé, Courbes de Weil semi-stables de discriminant une puissance $m$-ième. (French) [Semi-stable Weil curves of discriminant an $m$-th power]. J. Reine Angew. Math. 400 (1989), 173–184. Online from the Göttinger Digitalisierungszentrum via | MR 1013729 | Zbl 0693.14004

[30] S. J. Miller, Investigations of Zeros near the Central Point of Elliptic Curve $L$-functions. Exper. Math. 15 (2006), no. 3, 257–279. Preprint available at | MR 2264466 | Zbl 1131.11042

[31] P. Monsky, Mock Heegner points and congruent numbers. Math. Z. 204 (1990), no. 1, 45–67. Online via and possibly from springerlink.com via • P. Monsky, Three constructions of rational points on ${Y}^{2}={X}^{3}±NX$. Math. Z. 209 (1992), no. 3, 445–462. Available online from the Göttinger Digitalisierungszentrum via and also possibly from springerlink.com via ◇ P. Monsky, Errata: “Three constructions of rational points on ${Y}^{2}={X}^{3}±NX$.” Math. Z. 212 (1993), no. 1, 141. Corrects Lemma 4.7 and Theorem 4.8. Online from and also possibly available from springerlink.com via | Article | MR 1048066 | Zbl 0705.14023

[32] A. Pacetti, G. Tornaría, Computing central values of twisted L-series: the case of composite levels. To appear in Exper. Math. Preprint available at | MR 2484430

[33] PARI/GP, version 2.4.2-1896, Bordeaux, Oct. 2007. See

[34] H. Petersson, Konstrucktion der sämtlichen Lösungen einer Riemannschen Funktionalgleichung durch Dirichlet-Reihen mit Eulerscher Produktentwicklung. II. (German) [Construction of all the solution of a Riemannian functional equation by Dirichlet series with an Euler product development, II]. Math. Ann. 117 (1939), 39–64. Online from the Göttinger Digitalisierungszentrum via or possibly from springerlink.com via | Article | MR 1768 | Zbl 0022.12904

[35] G. Ricotta, T. Vidick, Hauteur asymptotique des points de Heegner. (French) [Asymptotic height of Heegner points]. To appear in Canadian Journal of Mathematics. Preprint online from | MR 2462452 | Zbl pre05382118

[36] N. F. Rogers, Rank Computations for the Congruent Number Elliptic Curves. Exper. Math. 9 (2000), no. 4, 591–594, | MR 1806294 | Zbl 1050.11061

[37] M. O. Rubinstein, University of Waterloo, data and code for $L$-functions, available online from in the directory DEGREE_2/ELLIPTIC/QUADRATIC_TWISTS/WEIGHT_THREE_HALVES/COEFFICIENTS

[38] G. Shimura, Introduction to the arithmetic theory of automorphic functions. Kanô Memorial Lectures, No. 1, Publications of the Math. Soc. of Japan, No. 11, Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, NJ, 1971. | MR 314766 | Zbl 0221.10029

[39] T. Shintani, On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J. 58 (1975), 83–126. Online from | MR 389772 | Zbl 0316.10016

[40] D. Simon, Computing the rank of elliptic curves over number fields. LMS J. Comput. Math. 5 (2002), 7–17. Online at and programme available from | MR 1916919 | Zbl 1067.11015

[41] N. C. Snaith, The derivative of $SO\left(2N+1\right)$ characteristic polynomials and rank 3 elliptic curves. In Ranks of Elliptic Curves and Random Matrix Theory, edited by J. B. Conrey, D. W. Farmer, F. Mezzadri, and N. C. Snaith, London Mathematical Society Lecture Note Series 341, Cambridge University Press (2007), 93–107. | MR 2322339 | Zbl pre05190706

[42] C.-J. de la Vallée Poussin, Recherches analytiques sur la théorie des nombres premiers. (French) [Analytic investigations in the theory of prime numbers]. Ann. Soc. scient. Bruxelles 20 (1896), 183–256.

[43] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier. (French) [On the Fourier coefficients of modular forms of half-integral weight]. J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484. | MR 646366 | Zbl 0431.10015

[44] M. Watkins, Real zeros of real odd Dirichlet L-functions, Math. Comp. 73 (2004), no. 245, 415–423. Online from ams.org via | Article | MR 2034130 | Zbl 1027.11063

[45] M. Watkins, Some remarks on Heegner point computations, notes from a short course at the Institut Henri Poincaré, see

[46] A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2) 141 (1995), no. 3, 443–551. Possibly online from jstor.org through | Article | MR 1333035 | Zbl 0823.11029