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15
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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Cited by 37 (11 self)
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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.
"Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical physics
 BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY
, 1993
"... Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and de ..."
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Cited by 24 (1 self)
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Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation.
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 14 (2 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.
Constructing elements in Shafarevich–Tate groups of modular motives
 in Number theory and algebraic geometry, London Mathematical Society Lecture Notes, Volume 303
, 2003
"... We study Shafarevich–Tate groups of motives attached to modular forms on Γ0(N) of weight> 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich–Tate groups, and give 16 examples in which a strong form of the Beilinson–Bloch conjecture would imply the existence of suc ..."
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Cited by 14 (1 self)
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We study Shafarevich–Tate groups of motives attached to modular forms on Γ0(N) of weight> 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich–Tate groups, and give 16 examples in which a strong form of the Beilinson–Bloch conjecture would imply the existence of such elements. We also use modular symbols and observations about Tamagawa numbers to compute nontrivial conjectural lower bounds on the orders of the Shafarevich–Tate groups of modular motives of low level and weight ≤ 12. Our methods build upon the idea of visibility due to Cremona and Mazur, but in the context of motives rather than abelian varieties. 1
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 10 (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
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
On deformation rings of residually reducible Galois representations and R = T theorems
, 2011
"... Abstract. We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation ρ0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually nonisomorphic constituents ρ1 and ρ2. Under some assumptions on Selm ..."
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Abstract. We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation ρ0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually nonisomorphic constituents ρ1 and ρ2. Under some assumptions on Selmer groups associated with ρ1 and ρ2 we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaïche and Chenevier we prove that I is principal for essentially selfdual representations and deduce statements about the structure of R. Using a new commutative algebra criterion we show that given enough information on the Hecke side one gets an R = Ttheorem. We then apply the technique to modularity problems for 2dimensional representations over an imaginary quadratic field and a 4dimensional representation over Q. 1.
Trivial zeros of PerrinRiou’s Lfunctions
, 2008
"... Abstract. In the previous paper [Ben2] we generalized Greenberg’s construction of the Linvariant to semistable padic representations. Here we prove that this construction is compatible with PerrinRiou’s theory of padic Lfunctions. Namely, using Nekováˇr’s machinery of Selmer complexes we prove ..."
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Abstract. In the previous paper [Ben2] we generalized Greenberg’s construction of the Linvariant to semistable padic representations. Here we prove that this construction is compatible with PerrinRiou’s theory of padic Lfunctions. Namely, using Nekováˇr’s machinery of Selmer complexes we prove that our Linvariant appears as an additional factor in the BlochKato type formula for special values of PerrinRiou’s Iwasawa Lfunction.
SELMER GROUPS AND CHOW GROUPS OF SELFPRODUCTS OF ALGEBRAIC VARIETIES
"... Abstract. Let X be a proper flat scheme over the ring of integers of a global field. We show that the Tate conjecture and the finiteness of the Chow group of vertical cycles on selfproducts of X implies the vanishing of the dual Selmer group of certain twists of tensor powers of representations occ ..."
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Abstract. Let X be a proper flat scheme over the ring of integers of a global field. We show that the Tate conjecture and the finiteness of the Chow group of vertical cycles on selfproducts of X implies the vanishing of the dual Selmer group of certain twists of tensor powers of representations occurring in the étale cohomology of X. 1.
Tamagawa numbers and . . .
"... The aim of the conference is to explain the formulation of the Tamagawa number conjecture by Bloch and Kato, and the proof of this conjecture for Dirichlet motives. The background necessary for the formulation of the conjecture is explained by the main speakers, in eight series of talks (A–H). Serie ..."
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The aim of the conference is to explain the formulation of the Tamagawa number conjecture by Bloch and Kato, and the proof of this conjecture for Dirichlet motives. The background necessary for the formulation of the conjecture is explained by the main speakers, in eight series of talks (A–H). Series Z consists of talks by the participants (model: Arbeitsgemeinschaft Oberwolfach – thus, the subjects of these talks have been fixed in advance; see the program below). These talks will provide certain details of the general theory entering the proof for Dirichlet motives. Preferably, the speakers will not be experts in the field. Graduate students and postdocs are particularly encouraged to participate, and to apply for one of the talks of the series Z. In order to do so, write an email to tamagawa@math.univparis13.fr, and establish a list of 3 talks (order of preference), each of which you would be ready to prepare (please do so before February 28). Instead of writing an email, you may use the registration form on the conference home page