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45
On the Equivariant Tamagawa Number Conjecture for Tate Motives, Part II
 DOCUMENTA MATH.
, 2006
"... Let K be any finite abelian extension of Q, k any subfield of K and r any integer. We complete the proof of the equivariant Tamagawa Number Conjecture for the pair (h0(Spec(K))(r), Z[Gal(K/k)]). ..."
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Cited by 38 (9 self)
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Let K be any finite abelian extension of Q, k any subfield of K and r any integer. We complete the proof of the equivariant Tamagawa Number Conjecture for the pair (h0(Spec(K))(r), Z[Gal(K/k)]).
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 34 (11 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
BlochKato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters
, 2002
"... The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the “special values ” of Lfunctions in terms of cohomological data. The main conjecture of Iwasawa theory describes a padic Lfunction in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of ..."
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Cited by 16 (3 self)
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The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the “special values ” of Lfunctions in terms of cohomological data. The main conjecture of Iwasawa theory describes a padic Lfunction in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of both conjectures (up to the prime 2) for Lfunctions attached to Dirichlet characters. We use the insight of Kato and B. PerrinRiou that these two conjectures can be seen as incarnations of the same mathematical content. In particular, they imply each other. By a bootstrapping process using the theory of Euler systems and explicit reciprocity laws, both conjectures are reduced to the analytic class number formula. Technical problems with primes dividing the order of the character are avoided by using the correct cohomological formulation of the main conjecture.
Equivariant epsilon constants, discriminants and Étale cohomology
 Proc. London Math. Soc
, 2001
"... this paper was written when the second named author 2 visited the Institute for Mathematics of the University of Augsburg in July 1999. He would like to thank Juergen Ritter for the warm hospitality shown him during this (and many other) visits ..."
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Cited by 14 (10 self)
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this paper was written when the second named author 2 visited the Institute for Mathematics of the University of Augsburg in July 1999. He would like to thank Juergen Ritter for the warm hospitality shown him during this (and many other) visits
On the Equivariant Tamagawa Number Conjecture for Abelian Extensions of a Quadratic Imaginary Field
 DOCUMENTA MATH.
, 2006
"... Let k be a quadratic imaginary field, p a prime which splits in k/Q and does not divide the class number hk of k. Let L denote a finite abelian extension of k and let K be a subextension of L/k. In this article we prove the ppart of the Equivariant Tamagawa Number Conjecture for the pair (h 0 (Spe ..."
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Cited by 13 (0 self)
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Let k be a quadratic imaginary field, p a prime which splits in k/Q and does not divide the class number hk of k. Let L denote a finite abelian extension of k and let K be a subextension of L/k. In this article we prove the ppart of the Equivariant Tamagawa Number Conjecture for the pair (h 0 (Spec(L)), Z[Gal(L/K)]).
Equivariant BlochKato conjecture and nonabelian Iwasawa main conjecture
 in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 149–162, Higher Ed
"... In this talk we explain the relation between the (equivariant) BlochKato conjecture for special values of Lfunctions and the Main Conjecture of (nonabelian) Iwasawa theory. On the way we will discuss briefly the case of Dirichlet characters in the abelian case. We will also discuss how “twisting ” ..."
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Cited by 11 (0 self)
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In this talk we explain the relation between the (equivariant) BlochKato conjecture for special values of Lfunctions and the Main Conjecture of (nonabelian) Iwasawa theory. On the way we will discuss briefly the case of Dirichlet characters in the abelian case. We will also discuss how “twisting ” in the nonabelian case would allow to reduce the general conjecture to the case of number fields. This is one the main motivations for a nonabelian Main
Equivariant Weierstrass Preparation and Values of Lfunctions at Negative Integers
 DOCUMENTA MATH.
, 2002
"... We apply an equivariant version of the padic Weierstrass Preparation Theorem in the context of possible noncommutative generalizations of the power series of Deligne and Ribet. We then consider CM abelian extensions of totally real fields and by combining our earlier considerations with the known ..."
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Cited by 11 (2 self)
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We apply an equivariant version of the padic Weierstrass Preparation Theorem in the context of possible noncommutative generalizations of the power series of Deligne and Ribet. We then consider CM abelian extensions of totally real fields and by combining our earlier considerations with the known validity of the Main Conjecture of Iwasawa theory we prove, modulo the conjectural vanishing of certain µinvariants, a (corrected version of a) conjecture of Snaith and the ‘rank zero component ’ of Kato’s Generalized Iwasawa Main Conjecture for Tate motives of strictly positive weight. We next use the validity of this case of Kato’s conjecture to prove a conjecture of Chinburg, Kolster, Pappas and Snaith and also to compute explicitly the Fitting ideals of certain natural étale cohomology groups in terms of the values of Dirichlet Lfunctions at negative integers. This computation improves upon results of Cornacchia and Østvær, of Kurihara and of Snaith, and, modulo the validity of a certain aspect of the QuillenLichtenbaum Conjecture, also verifies a finer and more general version of a well known conjecture of Coates and Sinnott.
Adjoint motives of modular forms and the Tamagawa number conjecture
, 2001
"... This paper concerns the Tamagawa number conjecture of Bloch and Kato [BK] for adjoint motives of modular forms of weight k ≥ 2. The conjecture relates the value at 0 of the associated Lfunction to arithmetic invariants of the motive. We prove that it holds up to powers of certain “bad primes. ” Th ..."
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Cited by 11 (2 self)
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This paper concerns the Tamagawa number conjecture of Bloch and Kato [BK] for adjoint motives of modular forms of weight k ≥ 2. The conjecture relates the value at 0 of the associated Lfunction to arithmetic invariants of the motive. We prove that it holds up to powers of certain “bad primes. ” The strategy for achieving
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 9 (3 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 ×
On the Values of Equivariant Zeta Functions of Curves over Finite Fields
 DOCUMENTA MATH.
, 2004
"... Let K/k be a finite Galois extension of global function fields of characteristic p. Let CK denote the smooth projective curve that has function field K and set G: = Gal(K/k). We conjecture a formula for the leading term in the Taylor expansion at zero of the Gequivariant truncated Artin Lfunctions ..."
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Cited by 7 (4 self)
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Let K/k be a finite Galois extension of global function fields of characteristic p. Let CK denote the smooth projective curve that has function field K and set G: = Gal(K/k). We conjecture a formula for the leading term in the Taylor expansion at zero of the Gequivariant truncated Artin Lfunctions of K/k in terms of the Weilétale cohomology of Gm on the corresponding open subschemes of CK. We then prove the ℓprimary component of this conjecture for all primes ℓ for which either ℓ ̸ = p or the relative algebraic Kgroup K0(Zℓ[G], Qℓ) is torsionfree. In the remainder of the manuscript we show that this result has the following consequences for K/k: if p ∤ G, then refined versions of all of Chinburg’s ‘ΩConjectures ’ in Galois module theory are valid; if the torsion subgroup of K × is a cohomologicallytrivial Gmodule, then Gross’s conjectural ‘refined class number formula ’ is valid; if K/k satisfies a certain natural classfield theoretical condition, then Tate’s recent refinement of Gross’s conjecture is valid.