A note on Greenberg’s conjecture and the abc conjecture

Ichimura, Humio

doi:10.1090/S0002-9939-98-04196-3

A note on Greenberg’s conjecture and the abc conjecture
HTML articles powered by AMS MathViewer

by Humio Ichimura PDF

Proc. Amer. Math. Soc. 126 (1998), 1315-1320 Request permission

Abstract:

For any totally real number field $k$ and any prime number $p$, Greenberg’s conjecture for $(k,p)$ asserts that the Iwasawa invariants $\lambda _p(k)$ and $\mu _p(k)$ are both zero. For a fixed real abelian field $k$, we prove that the conjecture is “affirmative” for infinitely many $p$ (which split in $k)$ if we assume the abc conjecture for $k$.

References

John Coates, $p$-adic $L$-functions and Iwasawa’s theory, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 269–353. MR 0460282
Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377–395. MR 528968, DOI 10.2307/1971116
Takashi Fukuda and Keiichi Komatsu, On $\textbf {Z}_p$-extensions of real quadratic fields, J. Math. Soc. Japan 38 (1986), no. 1, 95–102. MR 816225, DOI 10.2969/jmsj/03810095
Roland Gillard, Unités cyclotomiques, unités semi-locales et $\textbf {Z}_{l}$, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 1, xiv, 49–79 (French, with English summary). MR 526777
Ralph Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), no. 1, 263–284. MR 401702, DOI 10.2307/2373625
H. Ichimura and H. Sumida, On the Iwasawa invariants of certain real abelian fields II, International J. Math., 7 (1996), 721–744.
Kenkichi Iwasawa, On $\textbf {Z}_{l}$-extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246–326. MR 349627, DOI 10.2307/1970784
Serge Lang, Old and new conjectured Diophantine inequalities, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 1, 37–75. MR 1005184, DOI 10.1090/S0273-0979-1990-15899-9
Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
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