Computation of the zeros of 𝑝-adic 𝐿-functions

Ernvall, R.; Metsänkylä, T.

doi:10.1090/S0025-5718-1992-1122068-3

Computation of the zeros of $p$-adic $L$-functions
HTML articles powered by AMS MathViewer

by R. Ernvall and T. Metsänkylä PDF

Math. Comp. 58 (1992), 815-830 Request permission

Abstract:

The authors have computed the zeros of the Kubota-Leopoldt p-adic L-functions ${L_p}(s,\chi )$ for some small odd primes p and for a number of Dirichlet characters $\chi$. The zeros of the corresponding Iwasawa power series ${f_\theta }(T)$ are also computed. The characters $\chi$ (associated with quadratic extensions of the pth cyclotomic field) are chosen so as to cover as many different splitting types of ${f_\theta }(T)$ as possible. The $\lambda$-invariant of this power series, equal to its number of zeros, assumes values up to 8. The article is a report on these computations and their results, including the required theoretical background. Much effort is devoted to a study of the accuracy of the computed approximations.

References

A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
Nancy Childress and Robert Gold, Zeros of $p$-adic $L$-functions, Acta Arith. 48 (1987), no. 1, 63–71. MR 893462, DOI 10.4064/aa-48-1-63-71
R. Ernvall and T. Metsänkylä, A method for computing the Iwasawa $\lambda$-invariant, Math. Comp. 49 (1987), no. 179, 281–294. MR 890270, DOI 10.1090/S0025-5718-1987-0890270-X
Bruce Ferrero and Ralph Greenberg, On the behavior of $p$-adic $L$-functions at $s=0$, Invent. Math. 50 (1978/79), no. 1, 91–102. MR 516606, DOI 10.1007/BF01406470
Helmut Hasse, Zahlentheorie, Akademie-Verlag, Berlin, 1963 (German). Zweite erweiterte Auflage. MR 0153659

Cycles of reduced ideals in quadratic fields

Kenkichi Iwasawa and Charles C. Sims, Computation of invariants in the theory of cyclotomic fields, J. Math. Soc. Japan 18 (1966), 86–96. MR 202700, DOI 10.2969/jmsj/01810086

Calcul approché de la série d’Iwasawa pour les corps quadratiques

Tomio Kubota and Heinrich-Wolfgang Leopoldt, Eine $p$-adische Theorie der Zetawerte. I. Einführung der $p$-adischen Dirichletschen $L$-Funktionen, J. Reine Angew. Math. 214(215) (1964), 328–339 (German). MR 163900
Karin Lamprecht and Horst G. Zimmer, $p$-adic algorithms and the computation of zeros of $p$-adic $L$-functions, EUROCAL ’85, Vol. 2 (Linz, 1985) Lecture Notes in Comput. Sci., vol. 204, Springer, Berlin, 1985, pp. 491–502. MR 826581, DOI 10.1007/3-540-15984-3_{3}19

Zeros of p-adic L-functions and densities relating to Bernoulli numbers

Samuel S. Wagstaff Jr., Zeros of $p$-adic $L$-functions, Math. Comp. 29 (1975), no. 132, 1138–1143. MR 387253, DOI 10.1090/S0025-5718-1975-0387253-7
Samuel S. Wagstaff Jr., Zeros of $p$-adic $L$-functions. II, Number theory related to Fermat’s last theorem (Cambridge, Mass., 1981), Progr. Math., vol. 26, Birkhäuser, Boston, Mass., 1982, pp. 297–308. MR 685305
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
Lawrence C. Washington, Zeros of $p$-adic $L$-functions, Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981) Progr. Math., vol. 22, Birkhäuser, Boston, Mass., 1982, pp. 337–357. MR 693329