### 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 G_{m} 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 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.

Original language | English |
---|---|

Pages (from-to) | 555-626 |

Number of pages | 72 |

Journal | Documenta Mathematica |

Volume | 21 |

Issue number | 2016 |

Publication status | Published - 2016 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

_{m}

*Documenta Mathematica*,

*21*(2016), 555-626.

**On zeta elements for G _{m} .** / Burns, David; Kurihara, Masato; Sano, Takamichi.

Research output: Contribution to journal › Article

_{m}',

*Documenta Mathematica*, vol. 21, no. 2016, pp. 555-626.

_{m}Documenta Mathematica. 2016;21(2016):555-626.

}

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 -