Investigations of zeros near the central point of elliptic curve l-functions

Experimental Mathematics

Volumn 15, Issue 3, 2006, Pages 257-279

  Miller, Steven J ^a   Dueñez, Eduardo ^b  

^a BROWN UNIVERSITY   (United States)

^b UNIVERSITY OF TEXAS AT SAN ANTONIO   (United States)

Author keywords

Elliptic curves; Low lying zeros; N level statistics; Random matrix theory

Indexed keywords

EID: 33750870774     PISSN: 10586458     EISSN: 1944950X     Source Type: Journal    
DOI: 10.1080/10586458.2006.10128967     Document Type: Article

References (58)

1. S. Arms, A. Lozano-Robledo, and S. J. Miller. “Constructing Elliptic Curves over Q with Moderate Rank.” Preprint, to appear in J. Number Theory, 2006.
2. B. Birch and H. P. F. Swinnerton-Dyer. “Notes on Elliptic Curves, I.” J. Reine Angew. Math. 212 (1963), 7-25.
3. B. Birch and H. P. F. Swinnerton-Dyer. “Notes on Elliptic Curves, II.” J. Reine Angew. Math. 218 (1965), 79-108.
4. C. Breuil, B. Conrad, F. Diamond, and R. Taylor. “On the Modularity of Elliptic Curves over Q: Wild 3-adic Exercises.” J. Amer. Math. Soc. 14: 4 (2001), 843-939.
5. A. Brumer. “The Average Rank of Elliptic Curves, I.” Invent. Math. 109 (1992), 445-472.
6. A. Brumer and O. McGuinness. “The Behaviour of the Mordell-Weil Group of Elliptic Curves.” Bull. AMS 23 (1991), 375-382.
7. G. Casella and R. Berger., Statistical Inference. Second edition. Duxbury Advanced Series, 2002.
8. B. Conrey. “Notes on Eigenvalue Distribution for the Classical Compact Groups.” In Recent Perspectives in Random Matrix Theory and Number Theory, London Mathematical Society Lecture Note Series, 322, pp. 11-146. Cambridge: Cambridge University Press, 2005.
9. J. Cremona. Algorithms for Modular Elliptic Curves. Cambridge: Cambridge University Press, 1992.
10. C. David, J. Fearnley, and H. Kisilevsky. “On the Vanishing of Twisted L-Functions of Elliptic Curves.” Experiment. Math. 13: 2 (2004), 185-198.
11. P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou. “Uniform Asymptotics for Polynomials Orthogonal with Respect to Varying Exponential Weights and Applications to Universality Questions in Random Matrix Theory.” Comm. Pure Appl. Math. 52: 11 (1999), 1335-1425.
12. E. Dueñez. “Random Matrix Ensembles Associated to Compact Symmetric Spaces.” Commun. Math. Phys. 244 (2004), 29-61.
13. E. Dueñez and S. J. Miller. “The Low Lying Zeros of a GL(4) and a GL(6) Family of L Functions.” Preprint available online http://arxiv.org/pdf/math.NT/0506462, to appear in Compos. Math., 2006.
14. D. Farmer. “Modeling Families of L-Functions.” Preprint available online http://arxiv.org/pdf/math.NT/0511107, 2005.
15. S. Fermigier. “Zéros des fonctions L de courbes elliptiques.” Experimental Mathematics 1 (1992), 167-173.
16. S. Fermigier. “étude expérimentale du rang de familles de courbes elliptiques sur Q.” Experimental Mathematics 5 (1996), 119-130.
17. F. Gouvéa and B. Mazur. “The Square-Free Sieve and the Rank of Elliptic Curves.” J. Amer. Math. Soc. 4 (1991), 45-65.
18. D. Goldfeld. “The Class Number of Quadratic Fields and the Conjectures of Birch and Swinnerton-Dyer.” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3: 4 (1976), 624-663.
19. D. Goldfeld. “Conjectures on Elliptic Curves over Quadratic Fields.” In Number Theory (Proc. Conf. in Carbondale, 1979), pp. 108-118, Lecture Notes in Mathematics, 751. New York: Springer-Verlag, 1979.
20. B. Gross and D. Zagier. “Heegner Points and Derivatives of L-Series, II.” Math. Ann. 278 (1987), 497-562.
21. J. A. Harvey and D. J. Hughes. “Spacings of Nuclear Energy Levels.” Phys. Rev. 109: 2 (1958), 471-479.
22. R. U. Haq, A. Pandey, and O. Bohigas. “Fluctuation Properties of Nuclear Energy Levels: Do Theory and Experiment Agree?” Phys. Rev. Lett. 48 (1982), 1086-1089.
23. R. Heath-Brown. “The Average Analytic Rank of Elliptic Curves.” Duke Math. J. 122: 3 (2004), 591-623.
24. H. Helfgott. “On the Distribution of Root Numbers in Families of Elliptic Curves.” Preprint available online http://arxiv.org/pdf/math.NT/0408141, 2004.
25. D. Hejhal. “On the Triple Correlation of Zeros of the Zeta Function.” Internat. Math. Res. Notices 7 (1994), 294-302.
26. H. Iwaniec, W. Luo, and P. Sarnak. “Low Lying Zeros of Families of L-Functions.” Inst. Hautes ´ Etudes Sci. Publ. Math. 91 (2000), 55-131.
27. N. Katz and P. Sarnak., Random Matrices, Frobenius Eigenvalues and Monodromy, AMSColloquium Publications, 45. Providence: American Mathematical Society, 1999.
28. N. Katz and P. Sarnak. “Zeros of Zeta Functions and Symmetries.” Bull. AMS 36 (1999), 1-26.
29. J. P. Keating and N. C. Snaith. “Random Matrix Theory and ζ(1/2+it).” Comm. Math. Phys. 214: 1 (2000), 57-89.
30. J. P. Keating and N. C. Snaith. “Random Matrices and L-Functions.” Random Matrix Theory, J. Phys. A 36: 12 (2003), 2859-2881.
31. L. Mai. “The Analytic Rank of a Family of Elliptic Curves.” Canadian Journal of Mathematics 45 (1993), 847-862.
32. B. McKay. “The Expected Eigenvalue Distribution of a Large Regular Graph.” Linear Algebra Appl. 40 (1981), 203-216.
33. M. Mehta. Random Matrices. Second edition. Boston: Academic Press Inc., 1991.
34. J. Mestre. “Formules explicites et minorations de conducteurs de variétés algébriques.” Compositio Mathematica 58 (1986), 209-232.
35. P. Michel. “Rang moyen de familles de courbes elliptiques et lois de Sato-Tate.” Monat. Math. 120 (1995), 127-136.
36. S. J. Miller. “1-and 2-Level Densities for Families of Elliptic Curves: Evidence for the Underlying Group Symmetries.” PhD diss., Princeton University, 2002.
37. S. J. Miller. “1-and 2-Level Densities for Families of Elliptic Curves: Evidence for the Underlying Group Symmetries.” Compositio Mathematica 140: 4 (2004), 952-992.
38. S. J. Miller. “Variation in the Number of Points on Elliptic Curves and Applications to Excess Rank.” C. R. Math. Rep. Acad. Sci. Canada 27: 4 (2005), 111-120.
39. H. Montgomery. “The Pair Correlation of Zeros of the Zeta Function.” In Analytic Number Theory, pp. 181-193, Proceedings of Symposia in Pure Mathematics, 24. Providence, American Mathematical Society, 1973.
40. T. Nagao and M. Wadati. “Correlation Functions of Random Matrix Ensembles Related to Classical Orthogonal Polynomials.” J. Phys. Soc. Japan 60: 10 (1991), 3298-3322.
41. M. Rosen and J. Silverman. “On the Rank of an Elliptic Surface.” Invent. Math. 133 (1998), 43-67.
42. K. Rubin and A. Silverberg. “Rank Frequencies for Quadratic Twists of Elliptic Curves.” Experimental Mathematics 10: 4 (2001), 559-569.
43. M. Rubinstein. “Evidence for a Spectral Interpretation of the Zeros of L-Functions.” PhD diss., Princeton University, 1998.
44. M. Rubinstein. “Low-Lying Zeros of LFunctions and Random Matrix Theory.” Duke Math. J. 109 (2001), 147-181.
45. M. Rubinstein. “The L-Function Package.” Available online http://www.math.uwaterloo.ca/∼mrubinst, 2005.
46. Z. Rudnick and P. Sarnak. “Zeros of Principal L-Functions and Random Matrix Theory.” Duke Journal of Math. 81 (1996), 269-322.
47. J. Silverman. The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. New York: Springer-Verlag, 1986.
48. J. Silverman. Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 151. New York: Springer-Verlag, 1994.
49. J. Silverman. “The Average Rank of an Algebraic Family of Elliptic Curves.” J. Reine Angew. Math. 504 (1998), 227-236.
50. N. Snaith. “Derivatives of Random Matrix Characteristic Polynomials with Applications to Elliptic Curves.” J. Phys. A: Math. Gen. 38 (2005), 10345-10360.
51. C. Stewart and J. Top. “On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms.” Journal of the American Mathematical Society 40: 4 (1995), 943-973.
52. R. Taylor and A. Wiles. “Ring-Theoretic Properties of Certain Hecke Algebras.” Ann. Math. 141 (1995), 553-572.
53. C. Tracy and H. Widom. “Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices.” J. Statist. Phys. 92: 5, 6 (1998), 809-835.
54. M. Watkins. “Rank Distribution in a Family of Cubic Twists.” math. NT/0412427, 2004.
55. A. Wiles. “Modular Elliptic Curves and Fermat’s Last Theorem.” Ann. Math. 141 (1995), 443-551.
56. M. Young. “Lower Order Terms of the 1-Level Density of Families of Elliptic Curves.” IMRN 10 (2005), 587-633.
57. M. Young. “Low-Lying Zeros of Families of Elliptic Curves.” J. Amer. Math. Soc. 19: 1 (2006), 205-250.
58. D. Zagier and G. Kramarz. “Numerical Investigations Related to the L-Series of Certain Elliptic Curves.” J. Indian Math. Soc. 52 (1987), 51-69.