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

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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 of Kubota and Leopoldt, several *p*-adic places of the zero of were computed for the irregular primes .

**[1]**Z. I. BOREVICH (BOREVIČ) & I. R. SHAFAREVICH (ŠAFAREVIČ),*Number Theory*, "Nauka", Moscow, 1964; English transl., Pure and Appl. Math., vol. 20, Academic Press, New York, 1966. MR**30**#1080;**33**#4001. MR**0195803 (33:4001)****[2]**K. IWASAWA, "On some modules in the theory of cyclotomic fields,"*J. Math. Soc. Japan*, v. 16, 1964, pp. 42-82. MR**35**#6646. MR**0215811 (35:6646)****[3]**K. IWASAWA, "On*p*-adic*L*-functions,"*Ann. of Math.*(2), v. 89, 1969, pp. 198-205. MR**42**#4522. MR**0269627 (42:4522)****[4]**K. IWASAWA & C. C. SIMS, "Computation of invariants in the theory of cyclotomic fields,"*J. Math. Soc. Japan*, v. 18, 1966, pp. 86-96. MR**34**#2560. MR**0202700 (34:2560)****[5]**WELLS JOHNSON, "Irregular primes and cyclotomic invariants,"*Math. Comp.*, v. 29, 1975, pp. 113-120. MR**0376606 (51:12781)****[6]**V. V. KOBELEV, "Proof of Fermat's last theorem for all prime exponents less than 5500,"*Dokl. Akad. Nauk SSSR*, v. 190, 1970, pp. 767-768 =*Soviet Math. Dokl.*, v. 11, 1970, pp. 188-190. MR**41**#3363. MR**0258717 (41:3363)****[7]**T. KUBOTA & H. W. LEOPOLDT, "Eine*p*-adische Theorie der Zetawerte,"*J. Reine Angew. Math.*, v. 214/215, 1964, pp. 328-339. MR**29**#1199. MR**0163900 (29:1199)****[8]**D. H. LEHMER, E. LEHMER & H. S. VANDIVER, "An application of high-speed computing to Fermat's last theorem,"*Proc. Nat. Acad. Sci. U.S.A.*, v. 40, 1954, pp. 25-33. MR**15**, 778. MR**0061128 (15:778f)****[9]**J. L. SELFRIDGE, C. A. NICOL & H. S. VANDIVER, "Proof of Fermat's last theorem for all prime exponents less than 4002,"*Proc. Nat. Acad. Sci. U.S.A.*, v. 41, 1955, pp. 970-973. MR**17**, 348. MR**0072892 (17:348a)****[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.*, v. 40, 1954, pp. 732-735. MR**16**, 13. MR**0062758 (16:13f)**

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DOI:
https://doi.org/10.1090/S0025-5718-1975-0387253-7

Keywords:
*p*-adic *L*-functions,
cyclotomic field,
irregular primes

Article copyright:
© Copyright 1975
American Mathematical Society