Zeros of 𝑝-adic 𝐿-functions

Wagstaff, Samuel S.

doi:10.1090/S0025-5718-1975-0387253-7

Zeros of -adic -functions

Author: Samuel S. Wagstaff
Journal: Math. Comp. 29 (1975), 1138-1143
MSC: Primary 12B30
DOI: https://doi.org/10.1090/S0025-5718-1975-0387253-7
MathSciNet review: 0387253
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The p-adic coefficients and zeros of certain formal power series defined by Iwasawa have been calculated modulo various powers of p. Using these results and Iwasawa's formula for the p-adic L-function ${L_p}(s;\chi )$ of Kubota and Leopoldt, several p-adic places of the zero of ${L_p}(s;\chi )$ were computed for the irregular primes $p \leqslant 157$ .

[1] A. I. Borevich and I. R. Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. MR 0195803
[2] Kenkichi Iwasawa, On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan 16 (1964), 42–82. MR 0215811, https://doi.org/10.2969/jmsj/01610042
[3] Kenkichi Iwasawa, On 𝑝-adic 𝐿-functions, Ann. of Math. (2) 89 (1969), 198–205. MR 0269627, https://doi.org/10.2307/1970817
[4] Kenkichi Iwasawa and Charles C. Sims, Computation of invariants in the theory of cyclotomic fields, J. Math. Soc. Japan 18 (1966), 86–96. MR 0202700, https://doi.org/10.2969/jmsj/01810086
[5] Wells Johnson, Irregular primes and cyclotomic invariants, Math. Comp. 29 (1975), 113–120. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR 0376606, https://doi.org/10.1090/S0025-5718-1975-0376606-9
[6] V. V. Kobelev, A proof of Fermat’s theorem for all prime ewponents less that 5500., Dokl. Akad. Nauk SSSR 190 (1970), 767–768 (Russian). MR 0258717
[7] 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 0163900
[8] D. H. Lehmer, Emma Lehmer, and H. S. Vandiver, An application of high-speed computing to Fermat’s last theorem, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 25–33. MR 0061128
[9] J. L. Selfridge, C. A. Nicol, and H. S. Vandiver, Proof of Fermat’s last theorem for all prime exponents less than 4002, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 970–973. MR 0072892
[10] J. L. SELFRIDGE & B. W. POLLACK, "Fermat's last theorem is true for any exponent up to 25,000," Notices Amer. Math. Soc., v. 11, 1964, p. 97. Abstract #608-138.
[11] H. S. Vandiver, Examination of methods of attack on the second case of Fermat’s last theorem, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), 732–735. MR 0062758