Zeros of $p$-adic $L$-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

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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 $p \leqslant 157$.

- 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** - Kenkichi Iwasawa,
*On some modules in the theory of cyclotomic fields*, J. Math. Soc. Japan**16**(1964), 42–82. MR**215811**, DOI https://doi.org/10.2969/jmsj/01610042 - Kenkichi Iwasawa,
*On $p$-adic $L$-functions*, Ann. of Math. (2)**89**(1969), 198–205. MR**269627**, DOI https://doi.org/10.2307/1970817 - 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 https://doi.org/10.2969/jmsj/01810086 - Wells Johnson,
*Irregular primes and cyclotomic invariants*, Math. Comp.**29**(1975), 113–120. MR**376606**, DOI https://doi.org/10.1090/S0025-5718-1975-0376606-9 - 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** - 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** - 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**61128**, DOI https://doi.org/10.1073/pnas.40.1.25 - 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**72892**, DOI https://doi.org/10.1073/pnas.41.11.970
J. L. SELFRIDGE & B. W. POLLACK, "Fermat’s last theorem is true for any exponent up to 25,000," - 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**62758**, DOI https://doi.org/10.1073/pnas.40.8.732

*Notices Amer. Math. Soc.*, v. 11, 1964, p. 97. Abstract #608-138.

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Additional Information

Keywords:
<I>p</I>-adic <I>L</I>-functions,
cyclotomic field,
irregular primes

Article copyright:
© Copyright 1975
American Mathematical Society