On a conjecture of Coleman concerning Euler systems

Burns, David J.; Daoud, Alexandre; Seo, Soogil

doi:10.48550/arxiv.2104.09724

2021

DOI: 10.48550/arxiv.2104.09724

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On a conjecture of Coleman concerning Euler systems

David J. Burns¹,

Alexandre Daoud²,

Soogil Seo³

Abstract: We prove a distribution-theoretic conjecture of Robert Coleman, thereby also obtaining an explicit description of the complete set of Euler systems for the multiplicative group over Q.

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Dirichlet $L$-series at $s=0$ and the scarcity of Euler systems

Bullach¹,

Burns²,

Daoud³

et al. 2021

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Building on ideas of Coleman and Rubin, we develop a general theory of Euler systems 'over Z' for the multiplicative group over number fields. This theory has a range of concrete consequences including both the proof of a long-standing distribution-theoretic conjecture of Coleman and also an elementary interpretation of, and thereby a more direct approach to proving, the equivariant Tamagawa number conjecture (eTNC) for Dirichlet L-functions at s = 0. In this way we obtain an unconditional proof of the 'minus part' of the eTNC over CM extensions of totally real fields, an easier proof of the eTNC over Q, a proof of the eTNC over imaginary quadratic fields conditional only on a standard µ-vanishing hypothesis, and strong new evidence for the eTNC over general number fields. As a key part of this approach, we show that higher-rank Euler systems for a wide class of p-adic representations satisfy one divisibility in the natural main conjecture.

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Dirichlet $L$-series at $s=0$ and the scarcity of Euler systems

Bullach¹,

Burns²,

Daoud³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

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On a conjecture of Coleman concerning Euler systems

Abstract: We prove a distribution-theoretic conjecture of Robert Coleman, thereby also obtaining an explicit description of the complete set of Euler systems for the multiplicative group over Q.

Cited by 1 publication

References 11 publications

Dirichlet $L$-series at $s=0$ and the scarcity of Euler systems

Dirichlet $L$-series at $s=0$ and the scarcity of Euler systems

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