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Zeros of 2-adic  -functions and congruences
for class numbers and fundamental units
Authors:
Daniel C. Shanks, Patrick J. Sime and Lawrence C. Washington
Journal:
Math. Comp. 68 (1999), 1243-1255
MSC (1991):
Primary 11R11; Secondary 11S40
Published electronically:
February 10, 1999
MathSciNet review:
1622093
Full-text PDF Free Access
Abstract |
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Additional Information
Abstract: We study the imaginary quadratic fields such that the Iwasawa  -invariant equals 1, obtaining information on zeros of  -adic  -functions and relating this to congruences for fundamental units and class numbers.
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- P. T. Bateman, J. L. Selfridge, and S. S. Wagstaff, Jr., The new Mersenne conjecture, Am. Math. Monthly 96 (1989), 125-128. MR 90c:11009
- 2.
- N. Childress and R. Gold, Zeros of
 -adic  -functions, Acta Arith. 48 (1987), 63-71. MR 88i:11091
- 3.
- H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups of number fields, Number Theory, Noordwijkerhout 1983, 33-62, Springer Lecture Notes in Math. 1068 (1984). MR 85j:11144
- 4.
- B. Ferrero, The cyclotomic
 -extension of imaginary quadratic fields, Amer. J. Math. 102 (1980), 447-459. MR 81g:12006
- 5.
- P. Kaplan, Divisibilité par 8 du nombre des classes des corps quadratiques dont le 2-groupe des classes est cyclique, et réciprocité biquadratique, J. Math. Soc. Japan 25 (1973), 596-608. MR 48:2113
- 6.
- Y. Kida, On cyclotomic
 -extensions of imaginary quadratic fields, Tôhoku Math. J. (2), 31 (1979), 91-96. MR 80d:12003
- 7.
- F. Morain, e-mail announcement, April 29, 1996.
- 8.
- P. Morton, The quadratic number fields with cyclic 2-classgroups, Pacific J. Math. 108 (1983), 165-175. MR 84i:12001
- 9.
- C. D. Olds, Continued Fractions, Random House, New York, 1963. MR 26:3672
- 10.
- A. Pizer, On the 2-part of the class number of imaginary quadratic number fields, J. Number Theory 8 (1976), 184-192. MR 53:10759
- 11.
- L. Rédei and H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratisches Zahlkörpers, J. reine angew. Math. 170 (1934), 69-74.
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Additional Information
Daniel C. Shanks
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742
Patrick J. Sime
Affiliation:
Department of Mathematics & Comp. Sci., Caldwell College, Caldwell, NJ 07006
Email:
PSime@caldwell.edu
Lawrence C. Washington
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742
Email:
lcw@math.umd.edu
DOI:
http://dx.doi.org/10.1090/S0025-5718-99-01046-7
PII:
S 0025-5718(99)01046-7
Keywords:
Quadratic fields,
$p$-adic $L$-functions
Received by editor(s):
October 14, 1997
Published electronically:
February 10, 1999
Additional Notes:
The third author was partially supported by a grant from NSA, and also thanks the Institute for Advanced Study for its hospitality during part of the preparation of this paper.
Article copyright:
© Copyright 1999
American Mathematical Society
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