Perfecting the Nearly Perfect

Burns, David J.

doi:10.4310/pamq.2008.v4.n4.a3

“…In addition, the result of Corollary 1.8 is much stronger than (3) in that it shows the explicit structure of A(K) χ as a Galois module to be completely determined (conjecturally at least) by Stickelberger elements via the obvious (non-canonical…”

Section: Introductionmentioning

confidence: 94%

“…In this case there has as yet been no construction of a 'Weil-étale topology' for Y S := Spec(O K,S ) with all of the properties that are conjectured by Lichtenbaum in [34]. However, if Y S is a compactification of Y S and φ is the natural inclusion Y S ⊂ Y S , then the approach of [3] can be used to show that, should such a topology exist with all of the expected properties, then the groups H i c ((O K,S ) W , Z) defined above would be canonically isomorphic to the group [3,Rem. 3.8] and hence by the duality theorem in Weil-étale cohomology for curves over finite fields that is proved by Lichtenbaum in [33].…”

Section: Proposition 24 There Exist Canonical Complexes Ofmentioning

confidence: 99%

See 1 more Smart Citation

On zeta elements for $\mathbb{G}_{m}$

Burns¹,

Kurihara²,

Sano³

2014

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In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of 'zeta element', of 'Weil-étale cohomology complexes' and of 'integral Selmer groups' for the multiplicative group Gm over finite abelian extensions of number fields. We then conjecture a precise connection between zeta elements and Weil-étale cohomology complexes, we show this conjecture is equivalent to a special case of the equivariant Tamagawa number conjecture and we give an unconditional proof of the analogous statement for global function fields. We also show that the conjecture entails much detailed information about the arithmetic properties of generalized Stark elements including a new family of integral congruence relations between Rubin-Stark elements (that refines recent conjectures of Mazur and Rubin and of the third author) and explicit formulas in terms of these elements for the higher Fitting ideals of the integral Selmer groups of Gm, thereby obtaining a clear and very general approach to the theory of abelian Stark conjectures. As first applications of this approach, we derive, amongst other things, a proof of (a refinement of) a conjecture of Darmon concerning cyclotomic units, a proof of (a refinement of) Gross's 'Conjecture for Tori' in the case that the base field is Q, a proof of new cases of the equivariant Tamagawa number conjecture in situations in which the relevant p-adic L-functions have trivial zeroes, explicit conjectural formulas for both annihilating elements and, in certain cases, the higher Fitting ideals (and hence explicit structures) of ideal class groups, a reinterpretation of the p-adic Gross-Stark Conjecture in terms of the properties of zeta elements and a strong refinement of many previous results (of several authors) concerning abelian Stark conjectures.

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“…In this case there has as yet been no construction of a 'Weil-étale topology' for Y S := Spec(O K,S ) with all of the properties that are conjectured by Lichtenbaum in [34]. However, if Y S is a compactification of Y S and φ is the natural inclusion Y S ⊂ Y S , then the approach of [4] can be used to show that, should such a topology exist with all of the expected properties, then the groups H i c ((O K,S ) W , Z) defined above would be canonically isomorphic to the group [34]. (iii) The definition of RΓ T ((O K,S ) W , G m ) as the (shifted) linear dual of the complex RΓ c,T ((O K,S ) W , Z) is motivated by [4,Rem.…”

Section: 1mentioning

confidence: 99%

On zeta elements for $\Bbb G_m$

Burns¹,

Kurihara²,

Sano

³

2016

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In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of 'zeta element', of 'Weil-étale cohomology complexes' and of 'integral Selmer groups' for the multiplicative group G m over finite abelian extensions of number fields. We then conjecture a precise connection between zeta elements and Weil-étale cohomology complexes, we show this conjecture is equivalent to a special case of the equivariant Tamagawa number conjecture and we give an unconditional proof of the analogous statement for global function fields. We also show that the conjecture entails much detailed information about the arithmetic properties of generalized Stark elements including a new family of integral congruence relations between Rubin-Stark elements (that refines recent conjectures of Mazur and Rubin and of the third author) and explicit formulas in terms of these elements for the higher Fitting ideals of the integral Selmer groups of G m , thereby obtaining a clear and very general approach to the theory of abelian Stark conjectures. As first applications of this approach, we derive, amongst other things, a proof of (a refinement of) a conjecture of Darmon concerning cyclotomic units, a proof of (a refinement of) Gross's 'Conjecture for Tori' in the case that the base field is Q, explicit conjectural formulas for both annihilating elements and, in certain cases, the higher Fitting ideals (and hence explicit structures) of ideal class groups and a strong refinement of many previous results concerning abelian Stark conjectures. ContentsDavid Burns, Masato Kurihara, and Takamichi Sano 1.2. Refined class number formulas for G m 1.3. Selmer groups and their higher Fitting ideals 1.4. Galois structures of ideal class groups 1.5. Annihilators and Fitting ideals of class groups for small Σ 1.6. New verifications of the leading term conjecture 1.7. Notation 2. Canonical Selmer groups and complexes for G m 2.1. Integral dual Selmer groups 2.2. 'Weil-étale cohomology' complexes 2.3. Tate sequences 3. Zeta elements and the leading term conjecture 3.1. L-functions 3.2. The leading term lattice 3.3. Zeta elements 4. Preliminaries concerning exterior powers 4.1. Exterior powers 4.2. Rubin lattices 4.3. Homomorphisms between Rubin lattices 4.4. Congruences between exterior powers 5. Congruences for Rubin-Stark elements 5.1. The Rubin-Stark conjecture 5.2. Conventions for Rubin-Stark elements 5.3. Conjectures on Rubin-Stark elements 5.4. An explicit resolution 5.5. The equivalence of Conjectures 5.4 and 5.9 5.6. The leading term conjecture implies the Rubin-Stark conjecture 5.7. The leading term conjecture implies Conjecture 5.4 6. Conjectures of Darmon and of Gross 6.1. Darmon's Conjecture 6.2. Gross's conjecture for tori 7. Higher Fitting ideals of Selmer groups 7.1. Relative Fitting ideals 7.2. Statement of the conjecture 7.3. The leading term conjecture implies Conjecture 7.3 7.4. The proof of Theorem 1.10 7.5. The proof of Corollary 1.14 7.6. The higher relative Fitting ideals...

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“…where α X is constructed out of étale duality. Here the scheme X is of pure dimension d. This idea was suggested by works of Burns [8], Geisser [18] and the author [38]. The techniques involved in this paper rely on results due to Geisser and Levine on Bloch's cycle complex (see [17], [20], [21], [22] and [32]).…”

Section: R)mentioning

confidence: 99%

Zeta functions of regular arithmetic schemes at s=0

Morin¹

2014

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In [35] Lichtenbaum conjectured the existence of a Weil-étale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme X at s = 0 in terms of Euler-Poincaré characteristics. Assuming the (conjectured) finite generation of some étale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over Spec(Z). In particular, we obtain (unconditionally) the right Weil-étale cohomology for geometrically cellular schemes over number rings. We state a conjecture expressing the vanishing order and the special value up to sign of the Zeta function ζ(X , s) at s = 0 in terms of a perfect complex of abelian groups RΓW,c(X , Z). Then we relate this conjecture to Soulé's conjecture and to the Tamagawa number conjecture of Bloch-Kato, and deduce its validity in simple cases. (2010): 14F20 · 14G10 · 11S40 · 11G40 · 19F27 Mathematics Subject Classification

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Perfecting the Nearly Perfect

Cited by 6 publications

References 13 publications

On zeta elements for $\mathbb{G}_{m}$

On zeta elements for $\mathbb{G}_{m}$

On zeta elements for $\Bbb G_m$

Zeta functions of regular arithmetic schemes at s=0

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