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

top
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

top

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

top
  1. [1] J. COATES, p-adic L-functions and Iwasawa theory, Proc. Symp. Alg. Number theory, Durham (1975), 269-353. Zbl0393.12027
  2. [2] R. COLEMAN, Division values in local fields, Invent. Math., 53 (1979), 91-116. Zbl0429.12010MR81g:12017
  3. [3] R. COLEMAN, Local units modulo circular units, Proc. Amer. Math. Soc., 89, 1 (1983), 1-7. Zbl0528.12005MR85b:11088
  4. [4] L.J. FEDERER, Regulators, Iwasawa modules, and the Main conjecture for p = 2, in : N. KOBLITZ (ed.) : Number theory related to Fermat's Last Theorem, Birkhäuser Verlag (1982), 289-296. Zbl0504.12009
  5. [5] R. GILLARD, Unités cyclotomiques, unités semi-locales et Zl-extensions, Ann. Inst. Fourier, Grenoble, 29-1 (1979), 49-79. Zbl0387.12002
  6. [6] R. GOLD, J. KIM, Bases for cyclotomic units, Comp. Math., 71 (1989), 13-28. Zbl0687.12003MR90h:11101
  7. [7] G. GRAS, Sur l'annulation en 2 des classes relatives des corps abéliens, C.R. Math. Rep. Acad. Sci. Canada, 1 (1978), n°2, 107-110. Zbl0403.12005MR80k:12017
  8. [8] R. GREENBERG, On p-adic L-functions and cyclotomic fields I, Nagoya Math. J., 56 (1975), 61-77. Zbl0315.12008MR50 #12984
  9. [9] R. GREENBERG, On p-adic L-functions and cyclotomic fields II, Nagoya Math. J., 67 (1977), 139-158. Zbl0373.12007MR56 #2964
  10. [10] B. GROSS, p-adic L-series at s = 0, J. Math. Soc. Japan, 28 (1981), 979-994. Zbl0507.12010MR84b:12022
  11. [11] K. IWASAWA, On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan, 16 (1964), 42-82. Zbl0125.29207MR35 #6646
  12. [12] K. IWASAWA, Lectures on p-adic L-functions, Annals of Math. Studies n°74, Princeton University Press, Princeton 1972. Zbl0236.12001MR50 #12974
  13. [13] K. IWASAWA, On Zl-extensions of algebraic number fields, Ann. of Math., (2) (1979), 236-326. Zbl0285.12008
  14. [14] M. KOLSTER, A relation between the 2-primary parts of the main conjecture and the Birch-Tate conjecture, Canad. Math. Bull., 32 (1989), 248-251. Zbl0675.12004MR90k:11154
  15. [15] V. A. KOLYVAGIN, Euler systems. In : The Grothendieck Festschrift, vol. 2, 435-483, Birkhäuser Verlag 1990. Zbl0742.14017
  16. [16] S. LANG, Cyclotomic fields II, Graduate Texts in Mathematics, Springer Verlag, 1980. Zbl0435.12001MR81i:12004
  17. [17] B. MAZUR, A. WILES, Class fields of abelian extensions of Q, Invent. Math., 76 (1984), 179-330. Zbl0545.12005MR85m:11069
  18. [18] K. RUBIN, On the main conjecture of Iwasawa theory for imaginary quadratic fields, Invent. Math., 93 (1988), 701-713. Zbl0673.12004MR89j:11105
  19. [19] K. RUBIN, The Main Conjecture, Appendix to the second edition of S. Lang : Cyclotomic fields, Springer Verlag, 1990. 
  20. [20] W. SINNOTT, On the Stickelberger ideal and the circular units of an abelian field, Invent. Math., 62 (1980), 181-234. Zbl0465.12001MR82i:12004
  21. [21] W. SINNOTT, Appendix to L. Federer, B. Gross : Regulators and Iwasawa modules, Invent. Math., 62 (1981), 443-457. Zbl0468.12005
  22. [22] D. SOLOMON, On the class groups of imaginary abelian fields, Ann. Inst. Fourier, Grenoble, 40-3 (1990), 467-492. Zbl0694.12004MR92a:11133
  23. [23] L. WASHINGTON, Introduction to cyclotomic fields, Graduate Texts in Mathematics n°83, Springer Verlag, 1982. Zbl0484.12001MR85g:11001
  24. [24] A. WILES, The Iwasawa conjecture for totally real fields, Ann. Math., 131 (1990), 493-540. Zbl0719.11071MR91i:11163

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.