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Zeros of 2-adic $L$-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
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the imaginary quadratic fields such that the Iwasawa $\lambda _{2}$-invariant equals 1, obtaining information on zeros of $2$-adic $L$-functions and relating this to congruences for fundamental units and class numbers.

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  • 1. 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 $p$-adic $L$-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 $\mathbb{Z}_{2}$-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 $\mathbb{Z}_{2}$-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.
  • 12. L. Washington, Zeros of $p$-adic $L$-functions, Sém. Théorie des Nombres, Paris 1980-1981, Birkhäuser (1982), 337-357. MR 84f:12008
  • 13. L. Washington, Introduction to cyclotomic fields, Springer-Verlag, New York-Berlin, 1982. MR 85g:11001
  • 14. L. Washington, Siegel zeros for 2-adic $L$-functions, Number theory (Halifax, NS, 1994) CMS Conf. Proc., vol. 15, Amer. Math. Soc. (1995), 393-396. MR 96k:11145
  • 15. L. Washington, A family of cubic fields and zeros of 3-adic $L$-functions, J. Number Theory 63(1997), 408-417. MR 98e:11126

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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

Lawrence C. Washington
Affiliation: Department of Mathematics, University of Maryland, College Park, MD 20742

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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