Calculation of values of 𝐿-functions associated to elliptic curves

Akiyama, Shigeki; Tanigawa, Yoshio

doi:10.1090/S0025-5718-99-01051-0

Calculation of values of $L$-functions associated to elliptic curves
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by Shigeki Akiyama and Yoshio Tanigawa PDF

Math. Comp. 68 (1999), 1201-1231 Request permission

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

J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
F. Diamond, On deformation rings and Hecke rings, preprint.
H. M. Edwards, Riemann’s zeta function, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0466039
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, DOI 10.1080/10586458.1992.10504254
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
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 295536, DOI 10.1137/0708063
S. Hitotumatu, J. Yamauchi and T. Uno,Sûchikeisanhou III (Numerical Computing Methods III), Baihukan, 1971 (Japanese).
T. Kano (ed.) Riemann yosou (Riemann Hypothesis), Nihonhyouronsha, 1991 (Japanese).
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, DOI 10.1007/978-3-642-58018-5
Anthony W. Knapp, Elliptic curves, Mathematical Notes, vol. 40, Princeton University Press, Princeton, NJ, 1992. MR 1193029
L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0419394
A. F. Lavrik, Approximate functional equations of Dirichlet functions, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 134–185 (Russian). MR 0223313
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, DOI 10.1090/S0025-5718-1986-0829637-3
Ju. I. Manin, Cyclotomic fields and modular curves, Uspehi Mat. Nauk 26 (1971), no. 6(162), 7–71 (Russian). MR 0401653
Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66 (Russian). MR 0314846
A.M. Odlyzko, The $10^{20}$-th Zeros of the Riemann Zeta Function and $70$ Million of its Neighbors, preprint
A. P. Ogg, A remark on the Sato-Tate conjecture, Invent. Math. 9 (1969/70), 198–200. MR 258835, DOI 10.1007/BF01404324
Freydoon Shahidi, Symmetric power $L$-functions for $\textrm {GL}(2)$, Elliptic curves and related topics, CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994, pp. 159–182. MR 1260961, DOI 10.1090/crmp/004/11
Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
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
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
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
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
—, On calculations of zeros of various L-functions, Symposium on automorphic forms at Kinosaki (1993), 47-72.