The Minus Conjecture revisited

Greither, Cornélius; Kučera, Radan

doi:10.1515/crelle.2009.053

“…In this section, we formulate a refined version (see Conjecture 5.4) of these conjectures. We also recall a conjecture formulated by the first author, which was studied in [22], [2], [16], [17], [48], and [46] (see Conjecture 5.9). In [46,Th.…”

Section: Congruences For Rubin-stark Elementsmentioning

confidence: 92%

“…We next state a refinement of a conjecture that was formulated by the first author in [2] (the original version of which has been studied in many subsequent articles of different authors including [22], [16], [17], [48], and [46]).…”

Section: Conjectures On Rubin-stark Elementsmentioning

confidence: 99%

“…Then we have an equality [11], which was first proved by Mazur and Rubin in [36] via Kolyvagin systems. In §6, we shall use Corollary 5.16 to give a full proof of a refined version of Darmon's conjecture, and also give a new evidence for Gross's conjecture on tori [21], which was studied by Hayward [22], Greither and Kučera [16], [17].…”

Section: 4mentioning

confidence: 99%

“…Remark 6.6. The strongest previous evidence in favour of the conjecture for tori is that obtained by Greither and Kučera in [16,17], in which it is referred to as the 'Minus Conjecture' and studied in a slightly weaker form in order to remove any occurence of the auxiliary set T . More precisely, by using rather different methods they were able to prove that this conjecture was valid in the case that k = Q, K = F K + where F is imaginary quadratic of conductor f and class number h F and K + /Q is tamely ramified, abelian of exponent equal to an odd prime ℓ and ramified at precisely s primes {p i } 1≤i≤s each of which splits in F/Q; further, any of the following conditions are satisfied It is straightforward to show that the conjecture for tori implies their 'Minus conjecture', using [50, Chap.…”

Section: 4mentioning

confidence: 99%

See 2 more Smart Citations

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

Burns¹,

Kurihara²,

Sano³

2014

Preprint

0

View full text Add to dashboard Cite

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.

show abstract

“…In a forthcoming paper [4] we will explain how to prove (MC) for an arbitrary s under the assumption that l ≥ 3(s + 1) and l w F h F .…”

mentioning

confidence: 99%

On a conjecture concerning minus parts in the style of Gross

Greither

¹

,

Kučera

²

2008

Self Cite

View full text Add to dashboard Cite

On zeta elements for $\Bbb G_m$

Burns¹,

Kurihara²,

Sano

³

2016

View full text Add to dashboard Cite

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...

show abstract

The Minus Conjecture revisited

Cited by 3 publications

References 2 publications

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

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

On a conjecture concerning minus parts in the style of Gross

On zeta elements for $\Bbb G_m$

Contact Info

Product

Resources

About