## Tokyo Journal of Mathematics

### On Zeta Elements and Refined Abelian Stark Conjectures

Alice LIVINGSTONE BOOMLA

#### Abstract

We apply recent methods of Burns, Kurihara and Sano in [5] to study connections between the values at $s=0$ of the higher derivatives of abelian $L$-functions of number fields and the higher Fitting ideals of the canonical Selmer groups of $\mathbb{G}_m$. Whereas Burns, Kurihara and Sano apply these methods to the setting of the Rubin-Stark conjecture', we study the evaluators' defined in a more general setting by Emmons and Popescu in [7] and by Vallieres in [14]. This allows us to conjecture that the ideals formed from the images of the evaluators can be described precisely in terms of the higher Fitting ideals of the canonical Selmer groups of $\mathbb{G}_m$. Moreover, we are able to prove that this conjecture follows from the equivariant Tamagawa number conjecture.

#### Article information

Source
Tokyo J. Math., Volume 40, Number 1 (2017), 123-151.

Dates
First available in Project Euclid: 8 August 2017

https://projecteuclid.org/euclid.tjm/1502179219

Digital Object Identifier
doi:10.3836/tjm/1502179219

Mathematical Reviews number (MathSciNet)
MR3689982

Zentralblatt MATH identifier
06787091

#### Citation

LIVINGSTONE BOOMLA, Alice. On Zeta Elements and Refined Abelian Stark Conjectures. Tokyo J. Math. 40 (2017), no. 1, 123--151. doi:10.3836/tjm/1502179219. https://projecteuclid.org/euclid.tjm/1502179219

#### References

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