Notes on the dual of the ideal class groups of CM-fields
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.2, pp. 971-996.

In this paper, for a CM abelian extension K/k of number fields, we propose a conjecture which describes completely the Fitting ideal of the minus part of the Pontryagin dual of the T-ray class group of K for a set T of primes as a Gal(K/k)-module. Here, we emphasize that we consider the full class group, and do not throw away the ramifying primes (the object we study is not the quotient of the class group by the subgroup generated by the classes of ramifying primes). We prove that our conjecture is a consequence of the equivariant Tamagawa number conjecture, and also prove the Iwasawa theoretic version of our conjecture.

Dans cet article, pour une extension abélienne K/k de corps de nombres de type CM, nous proposons une conjecture qui décrit complètement l’idéal de Fitting de la partie moins du dual de Pontryagin du groupe de classes de rayon T de K, pour un ensemble T d’idéaux premiers, comme Gal(K/k)-module. Nous soulignons que nous considérons ici le groupe de classes au sens propre, sans laisser de côté les idéaux ramifiés (l’objet que nous étudions n’est pas le quotient du groupe de classes par le sous-groupe engendré par les classes des idéaux premiers ramifiés). Nous prouvons que notre conjecture est une conséquence de la conjecture de nombres de Tamagawa équivariante, et prouvons la version de notre conjecture en théorie d’Iwasawa.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1184
Classification: 11R29, 11R23, 11R37
Keywords: Class groups, Fitting ideals
Kurihara, Masato 1

1 Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Yokohama, 223-8522, Japan
@article{JTNB_2021__33_3.2_971_0,
     author = {Kurihara, Masato},
     title = {Notes on the dual of the ideal class groups of {CM-fields}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {971--996},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {3.2},
     year = {2021},
     doi = {10.5802/jtnb.1184},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1184/}
}
TY  - JOUR
AU  - Kurihara, Masato
TI  - Notes on the dual of the ideal class groups of CM-fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 971
EP  - 996
VL  - 33
IS  - 3.2
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1184/
DO  - 10.5802/jtnb.1184
LA  - en
ID  - JTNB_2021__33_3.2_971_0
ER  - 
%0 Journal Article
%A Kurihara, Masato
%T Notes on the dual of the ideal class groups of CM-fields
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 971-996
%V 33
%N 3.2
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1184/
%R 10.5802/jtnb.1184
%G en
%F JTNB_2021__33_3.2_971_0
Kurihara, Masato. Notes on the dual of the ideal class groups of CM-fields. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.2, pp. 971-996. doi : 10.5802/jtnb.1184. http://www.numdam.org/articles/10.5802/jtnb.1184/

[1] Burns, David An introduction to the equivariant Tamagawa number conjecture: the relation to Stark’s conjecture, Arithmetic of L-functions (IAS/Park City Mathematics Series), Volume 18, American Mathematical Society, 2011, pp. 127-152 | MR | Zbl

[2] Burns, David On derivatives of p-adic L-series at s=0, J. Reine Angew. Math., Volume 762 (2020), pp. 53-104 | DOI | MR | Zbl

[3] Burns, David; Greither, Cornelius Equivariant Weierstrass preparation and values of L-functions at negative integers, Doc. Math., Volume Extra Vol. (2003), pp. 157-185 | MR

[4] Burns, David; Greither, Cornelius On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math., Volume 153 (2003) no. 2, pp. 303-359 | DOI | MR | Zbl

[5] Burns, David; Kurihara, Masato; Sano, Takamichi On zeta elements for 𝔾 m , Doc. Math., Volume 21 (2016), pp. 555-626 | MR

[6] Burns, David; Kurihara, Masato; Sano, Takamichi On Iwasawa theory, zeta elements for 𝔾 m and the equivariant Tamagawa number conjecture, Algebra Number Theory, Volume 11 (2017) no. 7, pp. 1527-1571 | DOI | MR | Zbl

[7] Cassou-Nogues, Pierrette Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques, Invent. Math., Volume 51 (1979), pp. 29-59 | DOI | MR | Zbl

[8] Dasgupta, Samit; Kakde, Mahesh On the Brumer–Stark Conjecture (2020) (https://arxiv.org/abs/2010.00657)

[9] Deligne, Pierre; Ribet, Kenneth Values of abelian L-functions at negative integers over totally real fields, Invent. Math., Volume 59 (1979), pp. 227-286 | DOI | MR | Zbl

[10] Greither, Cornelius Computing Fitting ideals of Iwasawa modules, Math. Z., Volume 246 (2004) no. 4, pp. 733-767 | DOI | MR | Zbl

[11] Greither, Cornelius Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture, Compos. Math., Volume 143 (2007) no. 6, pp. 1399-1426 | DOI | MR | Zbl

[12] Greither, Cornelius; Kataoka, Takenori; Kurihara, Masato Fitting ideals of p-ramified Iwasawa modules over totally real fields (https://arxiv.org/abs/2006.05667, to appear in Selecta Math.)

[13] Greither, Cornelius; Kurihara, Masato Stickelberger elements, Fitting ideals of class groups of CM fields, and dualisation, Math. Z., Volume 260 (2008) no. 4, pp. 905-930 | DOI | MR | Zbl

[14] Greither, Cornelius; Kurihara, Masato; Tokio, Hibiki The second syzygy of the trivial G-module, and an equivariant main conjecture, Development of Iwasawa Theory – the Centennial of K. Iwasawa’s Birth (Advanced Studies in Pure Mathematics), Volume 86, Mathematical Society of Japan, 2020, pp. 317-349 | DOI | Zbl

[15] Greither, Cornelius; Popescu, Cristian An equivariant main conjecture in Iwasawa theory and applications, J. Algebr. Geom., Volume 24 (2015) no. 4, pp. 629-692 | DOI | MR | Zbl

[16] Gruenberg, Karl W.; Weiss, Alfred Galois invariants for local units, Q. J. Math, Volume 47 (1996) no. 185, pp. 25-39 | DOI | MR | Zbl

[17] Johnston, Henri; Nickel, Andreas An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications (2020) (https://arxiv.org/abs/2010.03186)

[18] Kurihara, Masato Iwasawa theory and Fitting ideals, J. Reine Angew. Math., Volume 561 (2003), pp. 39-86 | MR | Zbl

[19] Kurihara, Masato On the structure of ideal class groups of CM-fields, Doc. Math., Volume Extra Vol. (2003), pp. 539-563 | MR | Zbl

[20] Kurihara, Masato Rubin-Stark elements and ideal class groups, RIMS Kôkyûroku Bessatsu, Volume B53 (2015), pp. 343-363 | MR | Zbl

[21] Kurihara, Masato; Miura, Takashi Stickelberger ideals and Fitting ideals of class groups for abelian number fields, Math. Ann., Volume 350 (2011) no. 3, pp. 549-575 corrigendum in ibid. 374 (2019), no. 3-4, p. 2083-2088 | DOI | MR | Zbl

[22] Nickel, Andreas On non-abelian Stark-type conjectures, Ann. Inst. Fourier, Volume 61 (2011) no. 6, pp. 2577-2608 | DOI | Numdam | MR | Zbl

[23] Nickel, Andreas On the equivariant Tamagawa number conjecture in tame CM-extensions, Math. Z., Volume 268 (2011) no. 1-2, pp. 1-35 | DOI | MR | Zbl

[24] Ritter, Jürgen; Weiss, Alfred A Tate sequence for global units, Compos. Math., Volume 102 (1996) no. 2, pp. 147-178 | Numdam | MR | Zbl

[25] Siegel, Carl Ludwig Über die Fourierschen Koeffizienten von Modulformen, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., Volume 3 (1970), pp. 15-56 | Zbl

[26] Wiles, Andrew The Iwasawa conjecture for totally real fields, Ann. Math., Volume 131 (1990) no. 3, pp. 493-540 | DOI | MR | Zbl

Cited by Sources: