Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Calculation of values of $L$-functions
associated to elliptic curves

Authors: Shigeki Akiyama and Yoshio Tanigawa
Journal: Math. Comp. 68 (1999), 1201-1231
MSC (1991): Primary 11F11, 11G40, 11M26
Published electronically: February 10, 1999
MathSciNet review: 1627842
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We calculated numerically the values of $L$-functions of four typical elliptic curves in the critical strip in the range $\text{Im}(s)\leq 400$. We found that all the non-trivial zeros in this range lie on the critical line $\text{Re}(s)=1$ and are simple except the one at $s=1$. The method we employed in this paper is the approximate functional equation with incomplete gamma functions in the coefficients. For incomplete gamma functions, we continued them holomorphically to the right half plane $\text{Re}(s)>0$, which enables us to calculate for large $\text{Im}(s)$. Furthermore we remark that a relation exists between Sato-Tate conjecture and the generalized Riemann Hypothesis.

References [Enhancements On Off] (What's this?)

  • 1. J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
  • 2. F. Diamond, On deformation rings and Hecke rings, preprint.
  • 3. H. M. Edwards, Riemann’s zeta function, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 58. MR 0466039
  • 4. Stéfane Fermigier, Zéros des fonctions 𝐿 de courbes elliptiques, Experiment. Math. 1 (1992), no. 2, 167–173 (French, with English and French summaries). MR 1203872
  • 5. William B. Jones and Wolfgang J. Thron, Continued fractions, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. Analytic theory and applications; With a foreword by Felix E. Browder; With an introduction by Peter Henrici. MR 595864
  • 6. William B. Jones and W. J. Thron, A posteriori bounds for the truncation error of continued fractions, SIAM J. Numer. Anal. 8 (1971), 693–705. MR 0295536
  • 7. S. Hitotumatu, J. Yamauchi and T. Uno,Sûchikeisanhou III (Numerical Computing Methods III), Baihukan, 1971 (Japanese).
  • 8. T. Kano (ed.) Riemann yosou (Riemann Hypothesis), Nihonhyouronsha, 1991 (Japanese).
  • 9. Anatolij A. Karatsuba, Basic analytic number theory, Springer-Verlag, Berlin, 1993. Translated from the second (1983) Russian edition and with a preface by Melvyn B. Nathanson. MR 1215269
  • 10. Anthony W. Knapp, Elliptic curves, Mathematical Notes, vol. 40, Princeton University Press, Princeton, NJ, 1992. MR 1193029
  • 11. L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0419394
  • 12. A. F. Lavrik, Approximate functional equations of Dirichlet functions, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 134–185 (Russian). MR 0223313
  • 13. J. van de Lune, H. J. J. te Riele, and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. IV, Math. Comp. 46 (1986), no. 174, 667–681. MR 829637, 10.1090/S0025-5718-1986-0829637-3
  • 14. Ju. I. Manin, Cyclotomic fields and modular curves, Uspehi Mat. Nauk 26 (1971), no. 6(162), 7–71 (Russian). MR 0401653
  • 15. Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66 (Russian). MR 0314846
  • 16. A.M. Odlyzko, The $10^{20}$-th Zeros of the Riemann Zeta Function and $70$ Million of its Neighbors, preprint
  • 17. A. P. Ogg, A remark on the Sato-Tate conjecture, Invent. Math. 9 (1969/1970), 198–200. MR 0258835
  • 18. Freydoon Shahidi, Symmetric power 𝐿-functions for 𝐺𝐿(2), Elliptic curves and related topics, CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994, pp. 159–182. MR 1260961
  • 19. Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, 10.2307/2118560
  • 20. E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
  • 21. R. T. Turganaliev, An approximate functional equation and moments of the Dirichlet series generated by the Ramanujan function, Izv. Akad. Nauk Respub. Kazakhstan Ser. Fiz.-Mat. 5 (1992), 49–55 (Russian, with English, Russian and Kazakh summaries). MR 1254769
  • 22. H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948. MR 0025596
  • 23. Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, 10.2307/2118559
  • 24. Hiroyuki Yoshida, On calculations of zeros of 𝐿-functions related with Ramanujan’s discriminant function on the critical line, J. Ramanujan Math. Soc. 3 (1988), no. 1, 87–95. MR 975839
  • 25. -, On calculations of zeros of various L-functions, Symposium on automorphic forms at Kinosaki (1993), 47-72.

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 11F11, 11G40, 11M26

Retrieve articles in all journals with MSC (1991): 11F11, 11G40, 11M26

Additional Information

Shigeki Akiyama
Affiliation: Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, Niigata 950-2181, Japan

Yoshio Tanigawa
Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan

Keywords: Elliptic curve, $L$-function, approximate functional equation, Sato-Tate conjecture, Riemann Hypothesis
Received by editor(s): May 22, 1996
Received by editor(s) in revised form: December 11, 1996
Published electronically: February 10, 1999
Article copyright: © Copyright 1999 American Mathematical Society