Let G be a compact -adic Lie group, with no element of order , and having a closed normal subgroup H such that G/H is isomorphic to . We prove the existence of a canonical Ore set of non-zero divisors in the Iwasawa algebra of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to , we are able to define a characteristic element for every finitely generated -module M which has the property that the quotient of M by its -primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an elliptic curve E over , without complex multiplication, over the field F generated by the coordinates of all its p-power division points; here is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over .
@article{PMIHES_2005__101__163_0,
author = {Coates, John and Fukaya, Takako and Kato, Kazuya and Sujatha, Ramdorai and Venjakob, Otmar},
title = {The $GL_2$ main conjecture for elliptic curves without complex multiplication},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {163--208},
publisher = {Springer},
volume = {101},
year = {2005},
doi = {10.1007/s10240-004-0029-3},
zbl = {1108.11081},
language = {en},
url = {http://archive.numdam.org/articles/10.1007/s10240-004-0029-3/}
}
TY - JOUR AU - Coates, John AU - Fukaya, Takako AU - Kato, Kazuya AU - Sujatha, Ramdorai AU - Venjakob, Otmar TI - The $GL_2$ main conjecture for elliptic curves without complex multiplication JO - Publications Mathématiques de l'IHÉS PY - 2005 SP - 163 EP - 208 VL - 101 PB - Springer UR - http://archive.numdam.org/articles/10.1007/s10240-004-0029-3/ DO - 10.1007/s10240-004-0029-3 LA - en ID - PMIHES_2005__101__163_0 ER -
%0 Journal Article %A Coates, John %A Fukaya, Takako %A Kato, Kazuya %A Sujatha, Ramdorai %A Venjakob, Otmar %T The $GL_2$ main conjecture for elliptic curves without complex multiplication %J Publications Mathématiques de l'IHÉS %D 2005 %P 163-208 %V 101 %I Springer %U http://archive.numdam.org/articles/10.1007/s10240-004-0029-3/ %R 10.1007/s10240-004-0029-3 %G en %F PMIHES_2005__101__163_0
Coates, John; Fukaya, Takako; Kato, Kazuya; Sujatha, Ramdorai; Venjakob, Otmar. The $GL_2$ main conjecture for elliptic curves without complex multiplication. Publications Mathématiques de l'IHÉS, Tome 101 (2005), pp. 163-208. doi : 10.1007/s10240-004-0029-3. http://archive.numdam.org/articles/10.1007/s10240-004-0029-3/
1. and , Primeness, Semiprimeness and localization in Iwasawa algebras, preprint (2004). | MR
2. , Algebraic K-theory, Benjamin, New York (1968). | MR | Zbl
3. P. Balister, Congruences between special values of L-functions (unpublished) (1998).
4. N. Bourbaki, Commutative Algebra, Springer (1991).
5. , Pseudocompact algebras, profinite groups and class formations, J. Algebra 4 (1966), 442-470. | MR | Zbl
6. and , Tamagawa numbers for motives with (non-commutative) coefficients I, Doc. Math. 6 (2001), 501-570. | MR | Zbl
7. and , Tamagawa numbers for motives with (non-commutative) coefficients II, Am. J. Math. 125 (2003), 475-512. | MR
8. and , Euler characteristics and elliptic curves II, J. Math. Soc. Japan Proc. 53 (2001), 175-235. | MR | Zbl
9. , , and , Links between cyclotomic and GL2 Iwasawa theory, Doc. Math. Extra Volume: Kazuya Kato's 50th birthday (2003), 187-215.
10. , , and , Modules over Iwasawa algebras, J. Inst. Math. Jussieu 2 (2003), 73-108. | MR | Zbl
11. and , Euler-Poincaré characteristics of abelian varieties, CRAS 329, Série I (1999), 309-313. | MR | Zbl
12. and , Galois cohomology of elliptic curves, TIFR Lecture notes series, Narosa Publishing House (2000). | MR | Zbl
13. , Les constantes des équations fonctionnelles des fonctions L, Modular functions of one variable II, LNM 349, Springer (1973), 501-597. | MR | Zbl
14. , Valeurs de fonctions L et périodes d'intégrales, Proc. Sympos. Pure Math., XXXIII, Automorphic forms, representations and L-functions, Part 2, Amer. Math. Soc. (1979), 313-346. | Zbl
15. T. Dokchitser and V. Dokchitser, Numerical calculations in non-commutative Iwasawa theory, preprint (2004).
16. , Descent calculations for the elliptic curves of conductor 11, Proc. Lond. Math. Soc. 86 (2003), 583-606. | MR | Zbl
17. T. Fukaya and K. Kato, A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory, to appear in Proceedings of the St. Petersburg Mathematical Society. | MR
18. , On the structure of certain Galois groups, Invent. Math. 47 (1978), 85-99. | MR | Zbl
19. , Euler characteristics as invariants of Iwasawa modules, Proc. Lond. Math. Soc. 85 (2002), 634-658. | MR | Zbl
20. and , Equivariant Bloch-Kato conjecture and non-abelian Iwasawa main conjecture, Proceedings of the ICM, Vol. II (Beijing, 2002) (2002), 149-162. | MR | Zbl
21. , K1 of some non-commutative completed group rings, preprint (2004). | MR | Zbl
22. , Groupes analytiques p-adiques, Publ. Math., Inst. Hautes Étud. Sci. 26 (1965), 389-603. | Numdam | MR | Zbl
23. , Rational points of abelian varieties in towers of number fields, Invent. Math. 18 (1972), 183-266. | MR | Zbl
24. and , Noncommutative Noetherian Rings, Graduate Studies in Math. 30, AMS (1987). | MR | Zbl
25. , , and , Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften 323, Springer (2000). | MR | Zbl
26. , Groupes de Selmer d'une courbe elliptique à multiplication complexe, Compos. Math. 43 (1981), 387-417. | Numdam | Zbl
27. , The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), 25-68. | Zbl
28. , On the μ-invariant of p-adic L-functions, J. Number Theory 25 (1987), 20-33. | Zbl
29. , Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413-420, Oeuvres II, 264-271. | MR | Zbl
30. , Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331, Oeuvres III, 1-73. | Zbl
31. , Algèbre Locale, Multiplicités, 3rd ed., LNM 11, Springer (1975). | MR | Zbl
32. , Linear representations of finite groups, Graduate Texts in Mathematics 42 Springer (1977). | MR | Zbl
33. , Algebraic K-theory, LNM 76, Springer (1968). | MR | Zbl
34. , On stabilization for general linear groups over a ring, Math. USSR Sbornik 8 (1969), 383-400. | MR | Zbl
35. , On the Whitehead Determinant for Semi-local Rings, J. Algebra 283 (2005), 690-699. | MR | Zbl
36. , On the structure theory of the Iwasawa algebra of a p-adic Lie group, J. Eur. Math. Soc. 4 (2002), 271-311. | MR | Zbl
37. (), A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory, J. Reine Angew. Math. 559 (2003), 153-191. | MR | Zbl
38. , Characteristic elements in non-commutative Iwasawa theory, Habilitationschschrift, Heidelberg University (2003).
39. , Characteristic elements in non-commutative Iwasawa theory, to appear in J. Reine Angew. Math. | MR | Zbl
40. , On two variable p-adic L-functions, Ann. Math. 115 (1982), 411-449. | MR | Zbl
Cité par Sources :
