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1995 | 69 | 3 | 277-292
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The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields

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1. Introduction. Let k be a totally real number field. Let p be a fixed prime number and ℤₚ the ring of all p-adic integers. We denote by λ=λₚ(k), μ=μₚ(k) and ν=νₚ(k) the Iwasawa invariants of the cyclotomic ℤₚ-extension $k_∞$ of k for p (cf. [10]).
Then Greenberg's conjecture states that both λₚ(k) and μₚ(k) always vanish (cf. [8]). In other words, the order of the p-primary part of the ideal class group of kₙ remains bounded as n tends to infinity, where kₙ is the nth layer of $k_∞/k$. We know by the Ferrero-Washington theorem (cf. [2], [15]) that μₚ(k) always vanishes when k is an abelian (not necessarily totally real) number field. However, the conjecture remains unsolved up to now except for some special cases (cf. [1], [3], [5]-[8], [13]).
This paper is a continuation of our previous papers [3], [5]-[7] and [12], that is to say, we investigate Greenberg's conjecture when k is a real quadratic field and p is an odd prime number which splits in k. The purpose of this paper is to extend our previous results, and to give basic numerical data of k=ℚ(√m) for 0 ≤ m ≤ 10000 and p=3. On the basis of these data, we can verify Greenberg's conjecture for most of these k's.
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  • Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1, Shin-ei Narashino-shi, Chiba, 275 Japan
  • Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1, Okubo Shinjuku-ku, Tokyo, 169 Japan
  • [1] A. Candiotti, Computations of Iwasawa invariants and K₂, Compositio Math. 29 (1974), 89-111.
  • [2] B. Ferrero and L. C. Washington, The Iwasawa invariant μₚ vanishes for abelian number fields, Ann. of Math. 109 (1979), 377-395.
  • [3] T. Fukuda, Iwasawa λ-invariants of certain real quadratic fields, Proc. Japan Acad. 65A (1989), 260-262.
  • [4] T. Fukuda, Iwasawa λ-invariants of imaginary quadratic fields, J. College Industrial Technology Nihon Univ. 27 (1994), 35-88. (Corrigendum; to appear J. College Industrial Technology Nihon Univ.)
  • [5] T. Fukuda and K. Komatsu, On the λ invariants of ℤₚ-extensions of real quadratic fields, J. Number Theory 23 (1986), 238-242.
  • [6] T. Fukuda and K. Komatsu, On ℤₚ-extensions of real quadratic fields, J. Math. Soc. Japan 38 (1986), 95-102.
  • [7] T. Fukuda, K. Komatsu and H. Wada, A remark on the λ-invariants of real quadratic fields, Proc. Japan Acad. 62A (1986), 318-319.
  • [8] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263-284.
  • [9] R. Greenberg, On p-adic L-functions and cyclotomic fields II, Nagoya Math. J. 67 (1977), 139-158.
  • [10] K. Iwasawa, On $ℤ_l$-extensions of algebraic number fields, Ann. of Math. 98 (1973), 246-326.
  • [11] S. Mäki, The determination of units in real cyclic sextic fields, Lecture Notes in Math. 797, Springer, Berlin, 1980.
  • [12] H. Taya, On the Iwasawa λ-invariants of real quadratic fields, Tokyo J. Math. 16 (1993), 121-130.
  • [13] H. Taya, Computation of ℤ₃-invariants of real quadratic fields, preprint series, Waseda Univ. Technical Report No. 93-13, 1993.
  • [14] H. Wada and M. Saito, A table of ideal class groups of imaginary quadratic fields, Sophia Kôkyuroku in Math. 28, Depart. of Math., Sophia Univ. Tokyo, 1988.
  • [15] L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Math. 83, Springer, New York, 1982.
  • [16] H. Yokoi, On the class number of a relatively cyclic number field, Nagoya Math. J. 29 (1967), 31-44.
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