### Abstract

Let A be the inverse limit of the p-part of the ideal class groups in a ℤ^{r}_{p}-extension K_{∞}/K. Greenberg conjectures that if r is maximal, then A is pseudo-null as a module over the Iwasawa algebra A (that is, has codimension at least 2). We prove this conjecture in the case that K is the field of pth roots of unity, p has index of irregularity 1, satisfies Vandiver's conjecture, and satisfies a mild additional hypothesis on units. We also show that if K is the field of pth roots of unity and r is maximal, Greenberg's conjecture for K implies that the maximal p-ramified pro-p-extension of K cannot have a free pro-p quotient of rank r unless p is regular. Finally, we prove a generalization of a theorem of Iwasawa in the case r = 1 concerning the Kummer extension of K_{∞} generated by p-power roots of p-units. We show that the Galois group of this extension is torsion-free as a Λ-module if there is only one prime of K above p and K_{∞} contains all the p-power roots of unity.

Original language | English (US) |
---|---|

Pages (from-to) | 909-930 |

Number of pages | 22 |

Journal | American Journal of Mathematics |

Volume | 123 |

Issue number | 5 |

State | Published - Oct 2001 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

_{p}-extensions.

*American Journal of Mathematics*,

*123*(5), 909-930.

**Greenberg's conjecture and units in multiple ℤ _{p}-extensions.** / Mccallum, William G.

Research output: Contribution to journal › Article

_{p}-extensions',

*American Journal of Mathematics*, vol. 123, no. 5, pp. 909-930.

_{p}-extensions. American Journal of Mathematics. 2001 Oct;123(5):909-930.

}

TY - JOUR

T1 - Greenberg's conjecture and units in multiple ℤp-extensions

AU - Mccallum, William G

PY - 2001/10

Y1 - 2001/10

N2 - Let A be the inverse limit of the p-part of the ideal class groups in a ℤrp-extension K∞/K. Greenberg conjectures that if r is maximal, then A is pseudo-null as a module over the Iwasawa algebra A (that is, has codimension at least 2). We prove this conjecture in the case that K is the field of pth roots of unity, p has index of irregularity 1, satisfies Vandiver's conjecture, and satisfies a mild additional hypothesis on units. We also show that if K is the field of pth roots of unity and r is maximal, Greenberg's conjecture for K implies that the maximal p-ramified pro-p-extension of K cannot have a free pro-p quotient of rank r unless p is regular. Finally, we prove a generalization of a theorem of Iwasawa in the case r = 1 concerning the Kummer extension of K∞ generated by p-power roots of p-units. We show that the Galois group of this extension is torsion-free as a Λ-module if there is only one prime of K above p and K∞ contains all the p-power roots of unity.

AB - Let A be the inverse limit of the p-part of the ideal class groups in a ℤrp-extension K∞/K. Greenberg conjectures that if r is maximal, then A is pseudo-null as a module over the Iwasawa algebra A (that is, has codimension at least 2). We prove this conjecture in the case that K is the field of pth roots of unity, p has index of irregularity 1, satisfies Vandiver's conjecture, and satisfies a mild additional hypothesis on units. We also show that if K is the field of pth roots of unity and r is maximal, Greenberg's conjecture for K implies that the maximal p-ramified pro-p-extension of K cannot have a free pro-p quotient of rank r unless p is regular. Finally, we prove a generalization of a theorem of Iwasawa in the case r = 1 concerning the Kummer extension of K∞ generated by p-power roots of p-units. We show that the Galois group of this extension is torsion-free as a Λ-module if there is only one prime of K above p and K∞ contains all the p-power roots of unity.

UR - http://www.scopus.com/inward/record.url?scp=0035486398&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035486398&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0035486398

VL - 123

SP - 909

EP - 930

JO - American Journal of Mathematics

JF - American Journal of Mathematics

SN - 0002-9327

IS - 5

ER -