Zeros of 2-adic $L$-functions and congruences for class numbers and fundamental units
HTML articles powered by AMS MathViewer
- by Daniel C. Shanks, Patrick J. Sime and Lawrence C. Washington PDF
- Math. Comp. 68 (1999), 1243-1255 Request permission
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.References
- P. T. Bateman, J. L. Selfridge, and S. S. Wagstaff Jr., The new Mersenne conjecture, Amer. Math. Monthly 96 (1989), no. 2, 125–128. MR 992073, DOI 10.2307/2323195
- Nancy Childress and Robert Gold, Zeros of $p$-adic $L$-functions, Acta Arith. 48 (1987), no. 1, 63–71. MR 893462, DOI 10.4064/aa-48-1-63-71
- H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR 756082, DOI 10.1007/BFb0099440
- Bruce Ferrero, The cyclotomic $\textbf {Z}_{2}$-extension of imaginary quadratic fields, Amer. J. Math. 102 (1980), no. 3, 447–459. MR 573095, DOI 10.2307/2374108
- Pierre 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 (French). MR 323757, DOI 10.2969/jmsj/02540596
- Yûji Kida, On cyclotomic $\textbf {Z}_{2}$-extensions of imaginary quadratic fields, Tohoku Math. J. (2) 31 (1979), no. 1, 91–96. MR 526512, DOI 10.2748/tmj/1178229880
- F. Morain, e-mail announcement, April 29, 1996.
- Patrick Morton, The quadratic number fields with cyclic $2$-classgroups, Pacific J. Math. 108 (1983), no. 1, 165–175. MR 709708, DOI 10.2140/pjm.1983.108.165
- C. D. Olds, Continued fractions, Random House, New York, 1963. MR 0146146, DOI 10.5948/UPO9780883859261
- Arnold Pizer, On the $2$-part of the class number of imaginary quadratic number fields, J. Number Theory 8 (1976), no. 2, 184–192. MR 406975, DOI 10.1016/0022-314X(76)90100-1
- 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.
- Lawrence C. Washington, Zeros of $p$-adic $L$-functions, Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981) Progr. Math., vol. 22, Birkhäuser, Boston, Mass., 1982, pp. 337–357. MR 693329
- Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
- Lawrence C. Washington, Siegel zeros for $2$-adic $L$-functions, Number theory (Halifax, NS, 1994) CMS Conf. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 1995, pp. 393–396. MR 1353946
- Lawrence C. Washington, A family of cubic fields and zeros of $3$-adic $L$-functions, J. Number Theory 63 (1997), no. 2, 408–417. MR 1443772, DOI 10.1006/jnth.1997.2096
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
- 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.
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 68 (1999), 1243-1255
- MSC (1991): Primary 11R11; Secondary 11S40
- DOI: https://doi.org/10.1090/S0025-5718-99-01046-7
- MathSciNet review: 1622093