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

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

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]**B. Ferrero and L. C. Washington,*The Iwasawa invariant vanishes for abelian number fields*, Ann. of Math.**109**(1979), 377-395.MR**81a:12005****[Fu]**T. Fukuda,*On the vanishing of Iwasawa invariants of certain cyclic extensions of with prime degree*, Proc. Japan Acad.**73**A (1997), 108-110.MR**98i:11091****[Gr]**R. Greenberg,*On the Iwasawa invariants of totally real number fields*, Amer. J. Math.**98**(1976), 263-284. MR**53:5529****[Ich]**H. Ichimura,*A note on Greenberg's conjecture and the conjecture*, Proc. Amer. Math. Soc.**126**(1998), 1315-1320. MR**98j:11093****[IS]**H. Ichimura and H. Sumida,*On the Iwasawa -invariants of certain real abelian fields II*, International J. Math.**7**(1996), 721-744.MR**98e:11128c****[Iw1]**K. Iwasawa,*A note on class numbers of algebraic number fields*, Abh. Math. Sem. Univ. Hamburg**20**(1956), 257-258. MR**18:644d****[Iw2]**K. Iwasawa,*On -extensions of algebraic number fields*, Ann. of Math.**98**(1973), 246-326. MR**50:2120****[Iw3]**K. Iwasawa,*A note on capitulation problem for number fields II*, Proc. Japan Acad.**65**A (1989), 183-186. MR**90k:11139****[Kr]**J. S. Kraft,*Class numbers and Iwasawa invariants of quadratic fields*, Proc. Amer. Math. Soc.**124**(1996), 31-34.MR**96d:11112****[NH]**J. Nakagawa and K. Horie,*Elliptic curves with no rational points*, Proc. Amer. Math. Soc.**104**(1988), 20-24.MR**89k:11113****[Oz]**M. Ozaki,*The class group of -extensions over totally real number fields*, Tohoku Math. J.**49**(1997), 431-435. MR**98g:11123****[OT]**M. Ozaki and H. Taya,*On the Iwasawa -invariants of certain families of real quadratic fields*, Manuscripta Math.**94**(1997), 437-444. MR**99a:11122****[Ta]**H. Taya,*On cyclotomic -extensions of real quadratic fields*, Acta Arith.**74**(1996), 107-119.MR**97c:11101****[Wa]**L. C. Washington,*Introduction to Cyclotomic Fields*, Graduate Texts in Math. Vol. 83. Springer-Verlag, New York, Heidelberg, Berlin 1982.MR**85g:11001****[Ya]**G. Yamamoto,*On the vanishing of Iwasawa invariants of certain -extensions of*, Proc. Japan Acad.**73**A (1997), 45-47.MR**98b:11111**

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:
https://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