On pseudo-null Iwasawa modules

Kleine, Sören

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 2, pp. 583-618.

Nous étudions les sous-modules pseudo-nuls maximaux de certains modules d’Iwasawa construits à partir des groupes de classes d’idéaux dans des ${\mathbb{Z}}_p^k$-extensions de corps de nombres. Nous décrivons quelques critères de la non-trivialité de ces modules, en nous concentrant sur les cas $k=1$ et $k=2$. De plus, nous en déduisons des applications à des formes faibles de la conjecture de Greenberg généralisée (GGC).

We study the maximal pseudo-null submodules of Iwasawa modules arising from ideal class groups in ${\mathbb{Z}}_p^k$-extensions of number fields. We describe several sufficient criteria for the non-triviality of such modules, mainly in dimensions $k=1$ and $k=2$. This has applications to weak versions of Greenberg’s Generalised Conjecture (GGC).

Classification : 11R23
Mots-clés : Greenberg’s Generalised Conjecture, maximal pseudo-null submodules

[1] Badino, Raphaël; Nguyen Quang Do, Thong Sur les égalités du miroir et certaines formes faibles de la conjecture de Greenberg., Manuscr. Math., Volume 116 (2005) no. 3, pp. 323-340 | DOI | Zbl

[2] Bleher, Frauke M.; Chinburg, Ted; Greenberg, Ralph; Kakde, Mahesh; Pappas, Georgios; Sharifi, Romyar T.; Taylor, Martin J. Higher Chern classes in Iwasawa theory, Am. J. Math., Volume 142 (2020) no. 2, pp. 627-682 | DOI | MR

[3] Fujii, Satoshi Pseudo-null submodules of the unramified Iwasawa module for ${\mathbb{Z}}_p^2$-extensions., Interdiscip. Inf. Sci., Volume 16 (2010) no. 1, pp. 55-66 | DOI | MR | Zbl

[4] Fujii, Satoshi On restricted ramifications and pseudo-null submodules of Iwasawa modules for ${\mathbb{Z}}_p^2$-extensions., J. Ramanujan Math. Soc., Volume 29 (2014) no. 3, pp. 295-305 | MR | Zbl

[5] Fujii, Satoshi On Greenberg’s generalized conjecture for CM-fields, J. Reine Angew. Math., Volume 731 (2017), pp. 259-278 | DOI | MR | Zbl

[6] Fujii, Satoshi Some remarks on finite submodules of the unramified Iwasawa module of totally real fields, Proc. Japan Acad., Ser. A, Volume 96 (2020) no. 9, pp. 83-85 | DOI | MR | Zbl

[7] Fukuda, Takashi Remarks on ${\mathbb{Z}}_p$-extensions of number fields., Proc. Japan Acad., Ser. A, Volume 70 (1994) no. 8, pp. 264-266 | DOI | MR | Zbl

[8] Grandet, Marc; Jaulent, Jean-François Sur la capitulation dans une ${\mathbb{Z}}_{\ell }$-extension., J. Reine Angew. Math., Volume 362 (1985), pp. 213-217 | DOI | MR | Zbl

[9] Gras, Georges Classes généralisées invariantes., J. Math. Soc. Japan, Volume 46 (1994) no. 3, pp. 467-476 | DOI | Zbl

[10] Greenberg, Ralph The Iwasawa invariants of $\Gamma$-extensions of a fixed number field., Am. J. Math., Volume 95 (1973), pp. 204-214 | DOI | MR | Zbl

[11] Greenberg, Ralph Iwasawa theory – past and present., Class field theory – its centenary and prospect. Proceedings of the 7th MSJ International Research Institute of the Mathematical Society of Japan, Tokyo, Japan, June 3–12, 1998, Mathematical Society of Japan, 2001, pp. 335-385 | Zbl

[12] Iwasawa, Kenkichi On the $\mu$-invariants of ${\mathbb{Z}}_{\ell }$-extensions., Number theory, algebraic geometry and commutative algebra in honor of Yasuo Akizuki, Kinokuniya Book-Store Co., 1973, pp. 1-11 | Zbl

[13] Kleine, Sören A new approach for the investigation of Iwasawa invariants., Ph. D. Thesis, University of Göttingen (2015)

[14] Kleine, Sören Relative extensions of number fields and Greenberg’s generalised conjecture., Acta Arith., Volume 174 (2016) no. 4, pp. 367-392 | DOI | MR | Zbl

[15] Kleine, Sören Local behavior of Iwasawa’s invariants., Int. J. Number Theory, Volume 13 (2017) no. 4, pp. 1013-1036 | DOI | MR | Zbl

[16] Kleine, Sören $T$-ranks of Iwasawa modules., J. Number Theory, Volume 196 (2019), pp. 61-86 | DOI | MR | Zbl

[17] Lang, Serge Cyclotomic fields. I and II. With an appendix by Karl Rubin: The main conjecture. Combined 2nd edition., New York etc.: Springer-Verlag, 1990, xvii + 433 pages | Zbl

[18] The LMFDB Collaboration The L-functions and Modular Forms Database, 2021 (http://www.lmfdb.org)

[19] McCallum, William G.; Sharifi, Romyar T. A cup product in the Galois cohomology of number fields., Duke Math. J., Volume 120 (2003) no. 2, pp. 269-310 | DOI | MR | Zbl

[20] Minardi, John Iwasawa modules for ${\mathbb{Z}}_p^d$-extensions of algebraic number fields., Ph. D. Thesis, University of Washington (1986)

[21] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay Cohomology of number fields. 2nd ed., Springer, 2008, xv + 825 pages | DOI | Zbl

[22] Nguyen Quang Do, Thong Sur la conjecture faible de Greenberg dans le cas abélien $p$-décomposé., Int. J. Number Theory, Volume 2 (2006) no. 1, pp. 49-64 | DOI | Zbl

[23] Ozaki, Manabu The class group of ${\mathbb{Z}}_p$-extensions over totally real number fields., Tôhoku Math. J., Volume 49 (1997) no. 3, pp. 431-435 | DOI | MR | Zbl

[24] Ozaki, Manabu Iwasawa invariants of ${\mathbb{Z}}_p$-extensions over an imaginary quadratic field., Class field theory – its centenary and prospect. Proceedings of the 7th MSJ International Research Institute of the Mathematical Society of Japan, Tokyo, Japan, June 3–12, 1998, Mathematical Society of Japan, 2001, pp. 387-399 | MR | Zbl

[25] Ozaki, Manabu Construction of ${\mathbb{Z}}_p$-extensions with prescribed Iwasawa modules, J. Math. Soc. Japan, Volume 56 (2004) no. 3, pp. 787-801 | DOI | MR | Zbl

[26] The PARI Group PARI/GP version 2.11.2 (2021) (available from http://pari.math.u-bordeaux.fr/)

[27] Rosen, Michael Two theorems on Galois cohomology., Proc. Am. Math. Soc., Volume 17 (1966), pp. 1183-1185 | DOI | MR | Zbl

[28] Sands, Jonathan W. On small Iwasawa invariants and imaginary quadratic fields., Proc. Am. Math. Soc., Volume 112 (1991) no. 3, pp. 671-684 | DOI | MR | Zbl

[29] Schmithals, Bodo Kapitulation der Idealklassen und Einheitenstruktur in Zahlkörpern., J. Reine Angew. Math., Volume 358 (1985), pp. 43-60 | DOI | MR | Zbl

[30] Sharifi, Romyar T. On Galois groups of unramified pro-$p$ extensions, Math. Ann., Volume 342 (2008) no. 2, pp. 297-308 | DOI | MR | Zbl

[31] Washington, Lawrence C. Introduction to cyclotomic fields. 2nd ed., Springer, 1997, xiv + 487 pages | DOI | Zbl