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11
The GL2 main conjecture for elliptic curves without complex multiplication
 Publ. I.H.E.S. 101 (2005
"... The main conjectures of Iwasawa theory provide the only general method known at present for studying the mysterious relationship between purely arithmetic problems and the special values of complex Lfunctions, typified by the conjecture of Birch and Swinnerton ..."
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Cited by 28 (10 self)
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The main conjectures of Iwasawa theory provide the only general method known at present for studying the mysterious relationship between purely arithmetic problems and the special values of complex Lfunctions, typified by the conjecture of Birch and Swinnerton
Characteristic elements in noncommutative Iwasawa theory
, 2003
"... Let p be a prime number, which, for simplicity, we shall always assume odd. In the Iwasawa theory of an elliptic curve E over a number field k one has to distinguish between curves which do or do not admit complex multiplication (CM). For CMelliptic curves their deep arithmetic properties and the li ..."
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Cited by 7 (2 self)
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Let p be a prime number, which, for simplicity, we shall always assume odd. In the Iwasawa theory of an elliptic curve E over a number field k one has to distinguish between curves which do or do not admit complex multiplication (CM). For CMelliptic curves their deep arithmetic properties and the link between their Selmer group and special values of their HasseWeil Lfunctions are not only described by the (onevariable) main conjecture corresponding to the cyclotomic Zpextension kcyc of k, but also by the (twovariable) main conjecture corresponding to the extension k ∞ = k(Ep∞) which arises by adjoining the ppower division points Ep ∞ of E. Moreover, both conjectures are proven by Rubin [36] in the case that k is imaginary quadratic and E has CM by the ring of integers Ok of k. Also for nonCM elliptic curves one would like to at least formulate a main conjecture over the trivialzing extension k∞, but for lack of both an algebraic as well as analytic padic Lfunction this has not been achieved. The aim of this paper is to establish, under certain conditions, the existence of an algebraic padic Lfunction, viz as an element of the first Kgroup K1(ΛT) ∼ = Λ ×
Proof of the Main Conjecture of Noncommutative Iwasawa Theory for Totally Real Number Fields in Certain Cases
, 2008
"... Fix an odd prime p. Let G be a compact padic Lie group containing a closed, normal, prop subgroup H which is abelian and such that G/H is isomorphic to the additive group of padic integersZp. First we assume that H is finite and compute the Whitehead group of the Iwasawa algebra,Λ(G), of G. We al ..."
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Cited by 5 (0 self)
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Fix an odd prime p. Let G be a compact padic Lie group containing a closed, normal, prop subgroup H which is abelian and such that G/H is isomorphic to the additive group of padic integersZp. First we assume that H is finite and compute the Whitehead group of the Iwasawa algebra,Λ(G), of G. We also prove some results about certain localisation ofΛ(G) needed in Iwasawa theory. Let F be a totally real number field and let F ∞ be an admissible padic Lie extension of F with Galois group G. The computation of the Whitehead groups are used to show that the Main Conjecture for the extension F∞/F can be deduced from certain congruences between abelian padic zeta functions of Delige and Ribet. We prove these congruences with certain assumptions on G. This gives a proof of the Main Conjecture in many interesting cases such asZp⋊Zpextensions.
groups and noncommutative Iwasawa theory
 Jour. Alg. Geometry
"... Zeta isomorphisms and padic Lfunctions ..."
From classical to noncommutative Iwasawa theory  an introduction to the GL2 main conjecture
 4ECM Stockholm 2004, EMS
, 2005
"... This paper, which is an extended version of my talk ‘The GL2 main conjecture for elliptic curves without complex multiplication ’ given on the 4ECM, aims to give a survey on recent developments in noncommutative Iwasawa theory. It is written mainly ..."
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Cited by 2 (0 self)
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This paper, which is an extended version of my talk ‘The GL2 main conjecture for elliptic curves without complex multiplication ’ given on the 4ECM, aims to give a survey on recent developments in noncommutative Iwasawa theory. It is written mainly
On the Leading Terms of Zeta Isomorphisms and pAdic Lfunctions in NonCommutative Iwasawa Theory
 DOCUMENTA MATH.
, 2006
"... We discuss the formalism of Iwasawa theory descent in the setting of the localized K1groups of Fukaya and Kato. We then prove interpolation formulas for the ‘leading terms ’ of the global Zeta isomorphisms that are associated to certain Tate motives and of the padic Lfunctions that are associate ..."
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Cited by 1 (1 self)
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We discuss the formalism of Iwasawa theory descent in the setting of the localized K1groups of Fukaya and Kato. We then prove interpolation formulas for the ‘leading terms ’ of the global Zeta isomorphisms that are associated to certain Tate motives and of the padic Lfunctions that are associated to certain critical motives.
Iwasawa theory and . . .
"... We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture. ..."
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We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture.
FROM THE BIRCH & SWINNERTONDYER CONJECTURE OVER THE EQUIVARIANT TAMAGAWA NUMBER CONJECTURE TO NONCOMMUTATIVE IWASAWA THEORY A SURVEY
, 2005
"... This paper aims to give a survey on Fukaya and Kato’s article [19] which establishes the relation between the Equivariant Tamagawa Number Conjecture (ETNC) by Burns and Flach [7] and the noncommutative Iwasawa Main Conjecture (MC) (with padic Lfunction) as formulated by Coates, Fukaya, Kato, Sujat ..."
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This paper aims to give a survey on Fukaya and Kato’s article [19] which establishes the relation between the Equivariant Tamagawa Number Conjecture (ETNC) by Burns and Flach [7] and the noncommutative Iwasawa Main Conjecture (MC) (with padic Lfunction) as formulated by Coates, Fukaya, Kato, Sujatha and the author [11]. Moreover, we compare their approach with that of Huber and Kings [20] who formulate an Iwasawa Main Conjecture (without padic Lfunctions). We do not discuss these conjectures in full generality here, in fact we are mainly interested in the case of an abelian variety defined over Q. Nevertheless we formulate the conjectures for general motives over Q as far as possible. We follow closely the approach of Fukaya and Kato but our notation is sometimes inspired by [7, 20]. In particular, this article does not contain any new result, but hopefully serves as introduction to the original articles. See [37] for a more down to earth introduction to the GL2 Main Conjecture for an elliptic curve without complex multiplication. There we had pointed out that the Iwasawa main conjecture for an elliptic curve is morally the same as the (refined) Birch and Swinnerton Dyer (BSD) Conjecture for a whole tower of number fields. The work of Fukaya and
FROM THE BIRCH & SWINNERTONDYER CONJECTURE OVER THE EQUIVARIANT TAMAGAWA NUMBER CONJECTURE TO
, 2005
"... This paper aims to give a survey on Fukaya and Kato’s article [19] which establishes the relation between the Equivariant Tamagawa Number Conjecture (ETNC) by Burns and Flach [7] and the noncommutative Iwasawa Main Conjecture (MC) (with padic Lfunction) as formulated by Coates, Fukaya, Kato, Sujat ..."
Abstract
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This paper aims to give a survey on Fukaya and Kato’s article [19] which establishes the relation between the Equivariant Tamagawa Number Conjecture (ETNC) by Burns and Flach [7] and the noncommutative Iwasawa Main Conjecture (MC) (with padic Lfunction) as formulated by Coates, Fukaya, Kato, Sujatha and the author [11]. Moreover, we compare their approach with that of Huber and Kings [20] who formulate an Iwasawa Main Conjecture (without padic Lfunctions). We do not discuss these conjectures in full generality here, in fact we are mainly interested in the case of an abelian variety defined over Q. Nevertheless we formulate the conjectures for general motives over Q as far as possible. We follow closely the approach of Fukaya and Kato but our notation is sometimes inspired by [7, 20]. In particular, this article does not contain any new result, but hopefully serves as introduction to the original articles. See [37] for a more down to earth introduction to the GL2 Main Conjecture for an elliptic curve without complex multiplication. There we had pointed out that the Iwasawa main conjecture for an elliptic curve is morally the same as the (refined) Birch and Swinnerton Dyer (BSD) Conjecture for a whole tower of number fields. The work of Fukaya and