Euler system for Galois deformations

Tadashi Ochiai[1]

  • [1] Osaka University, Department of Mathematics, 1-16 Machikaneyama, Toyonaka, Osaka 560-0043 (Japon)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 1, page 113-146
  • ISSN: 0373-0956

Abstract

top
In this paper, we develop the Euler system theory for Galois deformations. By applying this theory to the Beilinson-Kato Euler system for Hida’s nearly ordinary modular deformations, we prove one of the inequalities predicted by the two-variable Iwasawa main conjecture. Our method of the proof of the Euler system theory is based on non-arithmetic specializations. This gives a new simplified proof of the inequality between the characteristic ideal of the Selmer group of a Galois deformation and the ideal associated to a Euler system even in the case of ℤ d p -extensions already treated by Kato, Perrin-Riou, Rubin.

How to cite

top

Ochiai, Tadashi. "Euler system for Galois deformations." Annales de l’institut Fourier 55.1 (2005): 113-146. <http://eudml.org/doc/116180>.

@article{Ochiai2005,
abstract = {In this paper, we develop the Euler system theory for Galois deformations. By applying this theory to the Beilinson-Kato Euler system for Hida’s nearly ordinary modular deformations, we prove one of the inequalities predicted by the two-variable Iwasawa main conjecture. Our method of the proof of the Euler system theory is based on non-arithmetic specializations. This gives a new simplified proof of the inequality between the characteristic ideal of the Selmer group of a Galois deformation and the ideal associated to a Euler system even in the case of $\mathbb \{Z\}^\{d_p\}$-extensions already treated by Kato, Perrin-Riou, Rubin.},
affiliation = {Osaka University, Department of Mathematics, 1-16 Machikaneyama, Toyonaka, Osaka 560-0043 (Japon)},
author = {Ochiai, Tadashi},
journal = {Annales de l’institut Fourier},
keywords = {Euler system; Hida theory; Iwasawa Main conjecture},
language = {eng},
number = {1},
pages = {113-146},
publisher = {Association des Annales de l'Institut Fourier},
title = {Euler system for Galois deformations},
url = {http://eudml.org/doc/116180},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Ochiai, Tadashi
TI - Euler system for Galois deformations
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 1
SP - 113
EP - 146
AB - In this paper, we develop the Euler system theory for Galois deformations. By applying this theory to the Beilinson-Kato Euler system for Hida’s nearly ordinary modular deformations, we prove one of the inequalities predicted by the two-variable Iwasawa main conjecture. Our method of the proof of the Euler system theory is based on non-arithmetic specializations. This gives a new simplified proof of the inequality between the characteristic ideal of the Selmer group of a Galois deformation and the ideal associated to a Euler system even in the case of $\mathbb {Z}^{d_p}$-extensions already treated by Kato, Perrin-Riou, Rubin.
LA - eng
KW - Euler system; Hida theory; Iwasawa Main conjecture
UR - http://eudml.org/doc/116180
ER -

References

top
  1. S. Bloch, K. Kato, L -functions and Tamagawa numbers of motives, in The Grothendieck Festschrift I, Progress in Math. 86 (1990), 333-400 Zbl0768.14001MR1086888
  2. N. Bourbaki, Eléments de mathématique, Algèbre commutative, Chapitre 5--7, (1985) Zbl0547.13002MR722608
  3. J.M. Fontaine (éd.), Périodes p -adiques, Séminaire de Bures (1988) 223 (1994) 
  4. P. Deligne, Formes modulaires et représentations ℓ -adiques, 179 (1969), Springer Verlag Zbl0206.49901
  5. P. Deligne, Valeurs des fonctions L et périodes d’intégrales, XXXIII, Part 2 (1979), 247-289, Amer. Math. Soc. Zbl0449.10022
  6. M. Flach, A generalisation of Cassels-Tate pairing, J. reine angew. Math. 412 (1990), 113-127 Zbl0711.14001MR1079004
  7. R. Greenberg, Iwasawa theory for p -adic representations, Advanced studies in Pure Math. 17 (1987), 97-137 Zbl0739.11045MR1097613
  8. R. Greenberg, Iwasawa theory for p -adic deformations of motives, Proceedings of Symposia in Pure Math. 55 (1994), 193-223 Zbl0819.11046MR1265554
  9. R. Greenberg, G. Stevens, p -adic L -functions and p -adic periods of modular forms, Invent. Math. 111 (1993), 407-447 Zbl0778.11034MR1198816
  10. H. Hida, Galois representations into GL 2 ( ℤ p [ [ X ] ] ) attached to ordinary cusp forms, Invent. Math. 85 (1986), 545-613 Zbl0612.10021MR848685
  11. H. Hida, Elementary theory of L -functions and Eisenstein series, 26 (1993), Cambridge University Press Zbl0942.11024MR1216135
  12. K. Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil L -functions via B dR , I, 1553 (1993), 50-163 Zbl0815.11051
  13. K. Kato, Series of lectures on Iwasawa main conjectures for modular elliptic curves, (1998) 
  14. K. Kato, p -adic Hodge theory and values of zeta functions of modular forms Zbl1142.11336MR2104361
  15. K. Kato, Euler systems, Iwasawa theory, and Selmer groups, Kodai Math. J. 22 (1999), 313-372 Zbl0993.11033MR1727298
  16. K. Kitagawa, On standard p -adic L -functions of families of elliptic cusp forms, in p -adic monodromy and the Birch and Swinnerton-Dyer conjecture, p.81-110, 165 (1994), Amer. Math. Soc. Zbl0841.11028MR1279604
  17. H. Matsumura, Commutative ring theory, 8 (1986), Cambridge University Press Zbl0603.13001MR879273
  18. B. Mazur, K. Rubin, Kolyvagin systems, (2001) Zbl1055.11041MR2031496
  19. B. Mazur, J. Tilouine, Représentations galoisiennes, différentielles de Kähler et ``conjectures principales", Inst. Hautes Études Sci. Publ. Math. 71 (1990), 65-103 Zbl0744.11053MR1079644
  20. B. Mazur, A. Wiles, Class fields of abelian extensions of ℚ , Invent. Math. 76 (1984), 179-330 Zbl0545.12005MR742853
  21. B. Mazur, A. Wiles, On p -adic analytic families of Galois representations, Compos. Math. 59 (1986), 231-264 Zbl0654.12008MR860140
  22. J.S. Milne, Arithmetic duality theorems,, (1986), Academic Press Zbl1127.14001MR881804
  23. J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields, 323 (2000), Springer-Verlag, Berlin Zbl0948.11001MR1737196
  24. T. Ochiai, Control theorem for Greenberg's Selmer groups for Galois deformations, J. Number Theory 88 (2001), 59-85 Zbl1090.11034MR1825991
  25. T. Ochiai, A generalization of the Coleman map for Hida deformations, Amer. J. Math. 125 (2003), 849-892 Zbl1057.11048MR1993743
  26. T. Ochiai, On the two-variable Iwasawa Main conjecture for Hida deformations Zbl1112.11051
  27. B. Perrin-Riou, Systèmes d’Euler p -adiques et théorie d’Iwasawa, Ann. Inst. Fourier 48 (1998), 1231-1307 Zbl0930.11078MR1662231
  28. K. Rubin, The ``main conjectures" of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), 25-68 Zbl0737.11030MR1079839
  29. K. Rubin, Euler systems, 147 (2000) Zbl0977.11001MR1749177
  30. J.-P. Serre, Cohomologie galoisienne, 5th ed., 5 (1994), Springer-Verlag Zbl0812.12002
  31. J. Tate, Relations between K 2 and Galois cohomology, Invent. Math. 36 (1976), 257-274 Zbl0359.12011MR429837
  32. A. Wiles, On λ -adic representations associated to modular forms, Invent. Math. 94 (1988), 529-573 Zbl0664.10013MR969243

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.