Class groups of abelian fields, and the main conjecture

Cornelius Greither

Annales de l'institut Fourier (1992)

  • Volume: 42, Issue: 3, page 449-499
  • ISSN: 0373-0956

Abstract

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This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case p = 2 , by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of χ -parts of p -class groups of abelian number fields: first for relative class groups of real fields (again including the case p = 2 ). As a consequence, a generalization of the Gras conjecture is stated and proved.

How to cite

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Greither, Cornelius. "Class groups of abelian fields, and the main conjecture." Annales de l'institut Fourier 42.3 (1992): 449-499. <http://eudml.org/doc/74963>.

@article{Greither1992,
abstract = {This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case $p=2$, by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of $\chi $-parts of $p$-class groups of abelian number fields: first for relative class groups of real fields (again including the case $p=2$). As a consequence, a generalization of the Gras conjecture is stated and proved.},
author = {Greither, Cornelius},
journal = {Annales de l'institut Fourier},
keywords = {cyclotomic extensions; p-adic L-functions; main conjecture of Iwasawa theory; units; abelian number fields; relative class groups; class groups of real fields; Gras conjecture},
language = {eng},
number = {3},
pages = {449-499},
publisher = {Association des Annales de l'Institut Fourier},
title = {Class groups of abelian fields, and the main conjecture},
url = {http://eudml.org/doc/74963},
volume = {42},
year = {1992},
}

TY - JOUR
AU - Greither, Cornelius
TI - Class groups of abelian fields, and the main conjecture
JO - Annales de l'institut Fourier
PY - 1992
PB - Association des Annales de l'Institut Fourier
VL - 42
IS - 3
SP - 449
EP - 499
AB - This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case $p=2$, by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of $\chi $-parts of $p$-class groups of abelian number fields: first for relative class groups of real fields (again including the case $p=2$). As a consequence, a generalization of the Gras conjecture is stated and proved.
LA - eng
KW - cyclotomic extensions; p-adic L-functions; main conjecture of Iwasawa theory; units; abelian number fields; relative class groups; class groups of real fields; Gras conjecture
UR - http://eudml.org/doc/74963
ER -

References

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  8. [8] R. GREENBERG, On p-adic L-functions and cyclotomic fields I, Nagoya Math. J., 56 (1975), 61-77. Zbl0315.12008MR50 #12984
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