## Tokyo Journal of Mathematics

### Fitting Ideals of Iwasawa Modules and of the Dual of Class Groups

#### Abstract

In this paper we study some problems related to a refinement of Iwasawa theory, especially questions about the Fitting ideals of several natural Iwasawa modules and of the dual of the class groups, as a sequel to our previous papers [8], [3]. Among other things, we prove that the annihilator of $\mathbb{Z}_{p}(1)$ times the Stickelberger element is not in the Fitting ideal of the dualized Iwasawa module if the $p$-component of the bottom Galois group is elementary $p$-abelian with $p$-rank $\geq 4$. Our results can be applied to the case that the base field is $\mathbb{Q}$.

#### Article information

Source
Tokyo J. of Math. Volume 39, Number 3 (2017), 619-642.

Dates
First available in Project Euclid: 6 October 2016

http://projecteuclid.org/euclid.tjm/1475723094

Digital Object Identifier
doi:10.3836/tjm/1475723094

#### Citation

GREITHER, Cornelius; KURIHARA, Masato. Fitting Ideals of Iwasawa Modules and of the Dual of Class Groups. Tokyo J. of Math. 39 (2017), no. 3, 619--642. doi:10.3836/tjm/1475723094. http://projecteuclid.org/euclid.tjm/1475723094.

#### References

• C. Greither, Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture, Compositio Math. 143 (2007), 1399–1426.
• C. Greither and R. Kučera, Annihilators of minus class groups of imaginary abelian fields, Ann. Inst. Fourier 57 (2007), 1623–1653.
• C. Greither and M. Kurihara, Tate sequences and Fitting ideals of Iwasawa modules, to appear in the “Vostokov volume,” Algebra i Analiz (St. Petersburg Math. J.) (2016).
• K. Iwasawa, On ${\bf Z}_{\ell}$-extensions of algebraic number fields, Ann. Math. 98 (1973), 246–326.
• K. Iwasawa, Riemann-Hurwitz formula and $p$-adic Galois representations for number fields, Tôhoku Math. J. 33 (1981), 263–288.
• T. Kimura, Algebraic class number formulae for cyclotomic fields (in Japanese), Sophia Kôkyûroku in Math. 22, Sophia University Tokyo, 1985.
• M. Kurihara, Iwasawa theory and Fitting ideals, J. reine angew. Math. 561 (2003), 39–86.
• M. Kurihara, On stronger versions of Brumer's conjecture, Tokyo J. Math. 34 (2011), 407–428.
• M. Kurihara and T. Miura, Stickelberger ideals and Fitting ideals of class groups for abelian number fields, Math. Annalen 350 (2011), 549–575.
• D. G. Northcott, Finite free resolutions, Cambridge Tracts in Math. 71, Cambridge Univ. Press, Cambridge New York, 1976.
• W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. Math. 108 (1978), 107–134.
• W. Sinnott, On the Stickelberger ideal and the circular units of an abelian field, Invent. Math. 62 (1980), 181–234.
• L. Washington, Introduction to cyclotomic fields, Graduate Texts in Math. 83, Springer-Verlag, 1982.
• A. Wiles, The Iwasawa conjecture for totally real fields, Ann. Math. 131 (1990), 493–540.