On zeta elements for Gm

David Burns, Masato Kurihara, Takamichi Sano

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

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-etale 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-etale 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, 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.

Original languageEnglish
Pages (from-to)555-626
Number of pages72
JournalDocumenta Mathematica
Volume21
Issue number2016
Publication statusPublished - 2016

Fingerprint

Étale Cohomology
Selmer Group
Refinement
Explicit Formula
Congruence Relation
Ideal Class Group
Cyclotomic
Function Fields
Gross
Number field
Equivariant
Thing
Multiplicative
Torus
Unit
Family

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Burns, D., Kurihara, M., & Sano, T. (2016). On zeta elements for Gm Documenta Mathematica, 21(2016), 555-626.

On zeta elements for Gm . / Burns, David; Kurihara, Masato; Sano, Takamichi.

In: Documenta Mathematica, Vol. 21, No. 2016, 2016, p. 555-626.

Research output: Contribution to journalArticle

Burns, D, Kurihara, M & Sano, T 2016, 'On zeta elements for Gm ', Documenta Mathematica, vol. 21, no. 2016, pp. 555-626.
Burns D, Kurihara M, Sano T. On zeta elements for Gm Documenta Mathematica. 2016;21(2016):555-626.
Burns, David ; Kurihara, Masato ; Sano, Takamichi. / On zeta elements for Gm In: Documenta Mathematica. 2016 ; Vol. 21, No. 2016. pp. 555-626.
@article{51162ad34eb447ce8336ee0926738c89,
title = "On zeta elements for Gm",
abstract = "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-etale 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-etale 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, 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.",
author = "David Burns and Masato Kurihara and Takamichi Sano",
year = "2016",
language = "English",
volume = "21",
pages = "555--626",
journal = "Documenta Mathematica",
issn = "1431-0635",
publisher = "Deutsche Mathematiker Vereinigung",
number = "2016",

}

TY - JOUR

T1 - On zeta elements for Gm

AU - Burns, David

AU - Kurihara, Masato

AU - Sano, Takamichi

PY - 2016

Y1 - 2016

N2 - 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-etale 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-etale 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, 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.

AB - 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-etale 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-etale 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, 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.

UR - http://www.scopus.com/inward/record.url?scp=85006021661&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85006021661&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85006021661

VL - 21

SP - 555

EP - 626

JO - Documenta Mathematica

JF - Documenta Mathematica

SN - 1431-0635

IS - 2016

ER -