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17
Tamagawa Numbers for Motives with (NonCommutative) Coefficients
 DOCUMENTA MATH.
, 2001
"... Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Qalgebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the Aequivariant Lfunction of M. This conjecture simultaneously generalizes a ..."
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Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple Qalgebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the Aequivariant Lfunction of M. This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, PerrinRiou et al. and also the central conjectures of classical Galois module theory as developed by Fröhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order A in A for which there exists a ‘projective Astructure ’ on M. The existence of such a structure is guaranteed if A is a maximal order, and also occurs in many natural examples where A is nonmaximal. In each such case the conjecture with respect to a nonmaximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in A by making use of the category of virtual objects introduced by Deligne.
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)]).
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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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
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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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
Symmetric Square LFunctions and ShafarevichTate Groups.” Experiment
 Math
"... We use Zagier's method to compute the critical values of the 1. Introduction symmetric square Lfunctions of six cuspidal eigenforms of level 2. Calculating the Critical Values one with rational coefficients. According to the BlochKato 3. Tables of Results conjecture, certain large primes divi ..."
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Cited by 6 (3 self)
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We use Zagier's method to compute the critical values of the 1. Introduction symmetric square Lfunctions of six cuspidal eigenforms of level 2. Calculating the Critical Values one with rational coefficients. According to the BlochKato 3. Tables of Results conjecture, certain large primes dividing these critical values
On the equivariant Tamagawa number conjecture for CM elliptic curves
 Proceedings of the Symposium on Algebraic Number Theory and Related Topics, RIMS Kôkyûroku Bessatsu, B4
, 2007
"... Abstract. In this article we consider the equivariant Tamagawa number conjecture for Hecke characters over imaginary quadratic fields. Assuming weak Leopoldt conjecture and the equivariant main conjecture for imaginary quadratic fields, we give a proof of a weak version of the equivariant Tamagawa ..."
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Abstract. In this article we consider the equivariant Tamagawa number conjecture for Hecke characters over imaginary quadratic fields. Assuming weak Leopoldt conjecture and the equivariant main conjecture for imaginary quadratic fields, we give a proof of a weak version of the equivariant Tamagawa number conjecture for Hecke characters. 1.
A Bound for the Torsion in the KTheory of Algebraic Integers
 DOCUMENTA MATH.
, 2003
"... Let m be an integer bigger than one, A a ring of algebraic integers, F its fraction field, and Km(A) the mth Quillen Kgroup of A. We give a (huge) explicit bound for the order of the torsion subgroup of Km(A) (up to small primes), in terms of m, the degree of F over Q, and its absolute discrimin ..."
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Let m be an integer bigger than one, A a ring of algebraic integers, F its fraction field, and Km(A) the mth Quillen Kgroup of A. We give a (huge) explicit bound for the order of the torsion subgroup of Km(A) (up to small primes), in terms of m, the degree of F over Q, and its absolute discriminant.
Generators and relations for K2OF , F imaginary quadratic, in preparation
"... Abstract. Tate’s algorithm [31] for computing K2OF for rings of integers in a number field has been adapted for the computer and gives explicit generators for the group and sharp bounds on their order—the latter, together with some structural results on the pth primary part of K2OF due to Tate and ..."
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Abstract. Tate’s algorithm [31] for computing K2OF for rings of integers in a number field has been adapted for the computer and gives explicit generators for the group and sharp bounds on their order—the latter, together with some structural results on the pth primary part of K2OF due to Tate and Keune, gives a proof of its structure for many imaginary quadratic fields, confirming earlier conjectural results in [7].
GENERATORS AND RELATIONS FOR K2OF
"... Abstract. Tate’s algorithm for computing K2OF for rings of integers in a number field has been adapted for the computer and gives explicit generators for the group and sharp bounds on their order—the latter, together with some structural results on the pprimary part of K2OF due to Tate and Keune, g ..."
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Abstract. Tate’s algorithm for computing K2OF for rings of integers in a number field has been adapted for the computer and gives explicit generators for the group and sharp bounds on their order—the latter, together with some structural results on the pprimary part of K2OF due to Tate and Keune, gives a proof of its structure for many number fields of small discriminants, confirming earlier conjectural results. For the first time, tame kernels of non