Iwasawa invariants and class numbers

of quadratic fields for the prime

Author:
Hisao Taya

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1285-1292

MSC (1991):
Primary 11R23, 11R11, 11R29

Published electronically:
August 3, 1999

MathSciNet review:
1641133

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a square-free integer with and . Put and . For the cyclotomic -extension of , we denote by the -th layer of over . We prove that the -Sylow subgroup of the ideal class group of is trivial for all integers if and only if the class number of is not divisible by the prime . This enables us to show that there exist infinitely many real quadratic fields in which splits and whose Iwasawa -invariant vanishes.

**[FW]**Bruce Ferrero and Lawrence C. Washington,*The Iwasawa invariant 𝜇_{𝑝} vanishes for abelian number fields*, Ann. of Math. (2)**109**(1979), no. 2, 377–395. MR**528968**, 10.2307/1971116**[Fu]**Takashi Fukuda,*On the vanishing of Iwasawa invariants of certain cyclic extensions of 𝐐 with prime degree*, Proc. Japan Acad. Ser. A Math. Sci.**73**(1997), no. 6, 108–110. MR**1469684****[Gr]**Ralph Greenberg,*On the Iwasawa invariants of totally real number fields*, Amer. J. Math.**98**(1976), no. 1, 263–284. MR**0401702****[Ich]**Humio Ichimura,*A note on Greenberg’s conjecture and the 𝑎𝑏𝑐 conjecture*, Proc. Amer. Math. Soc.**126**(1998), no. 5, 1315–1320. MR**1443156**, 10.1090/S0002-9939-98-04196-3**[IS]**Humio Ichimura and Hiroki Sumida,*On the Iwasawa invariants of certain real abelian fields. II*, Internat. J. Math.**7**(1996), no. 6, 721–744. MR**1417782**, 10.1142/S0129167X96000384**[Iw1]**Kenkichi Iwasawa,*A note on class numbers of algebraic number fields*, Abh. Math. Sem. Univ. Hamburg**20**(1956), 257–258. MR**0083013****[Iw2]**Kenkichi Iwasawa,*On 𝑍_{𝑙}-extensions of algebraic number fields*, Ann. of Math. (2)**98**(1973), 246–326. MR**0349627****[Iw3]**Kenkichi Iwasawa,*A note on capitulation problem for number fields. II*, Proc. Japan Acad. Ser. A Math. Sci.**65**(1989), no. 6, 183–186. MR**1011867****[Kr]**James S. Kraft,*Class numbers and Iwasawa invariants of quadratic fields*, Proc. Amer. Math. Soc.**124**(1996), no. 1, 31–34. MR**1301510**, 10.1090/S0002-9939-96-03085-7**[NH]**Jin Nakagawa and Kuniaki Horie,*Elliptic curves with no rational points*, Proc. Amer. Math. Soc.**104**(1988), no. 1, 20–24. MR**958035**, 10.1090/S0002-9939-1988-0958035-0**[Oz]**Manabu Ozaki,*The class group of 𝑍_{𝑝}-extensions over totally real number fields*, Tohoku Math. J. (2)**49**(1997), no. 3, 431–435. MR**1464188**, 10.2748/tmj/1178225114**[OT]**Manabu Ozaki and Hisao Taya,*On the Iwasawa 𝜆₂-invariants of certain families of real quadratic fields*, Manuscripta Math.**94**(1997), no. 4, 437–444. MR**1484637**, 10.1007/BF02677865**[Ta]**Hisao Taya,*On cyclotomic 𝑍_{𝑝}-extensions of real quadratic fields*, Acta Arith.**74**(1996), no. 2, 107–119. MR**1373702****[Wa]**Lawrence C. Washington,*Introduction to cyclotomic fields*, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR**718674****[Ya]**Gen Yamamoto,*On the vanishing of Iwasawa invariants of certain (𝑝,𝑝)-extensions of 𝑄*, Proc. Japan Acad. Ser. A Math. Sci.**73**(1997), no. 3, 45–47. MR**1453528**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
11R23,
11R11,
11R29

Retrieve articles in all journals with MSC (1991): 11R23, 11R11, 11R29

Additional Information

**Hisao Taya**

Email:
taya@math.is.tohoku.ac.jp

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05177-1

Keywords:
Iwasawa invariants,
real quadratic fields,
class numbers

Received by editor(s):
August 27, 1997

Received by editor(s) in revised form:
June 22, 1998

Published electronically:
August 3, 1999

Additional Notes:
This research was partially supported by the Grant-in-Aid for Encouragement of Young Scientists, The Ministry of Education, Science, Sports and Culture, Japan.

Dedicated:
Dedicated to Professor Koji Uchida on his 60th birthday

Communicated by:
David E. Rohrlich

Article copyright:
© Copyright 2000
American Mathematical Society