On Zeta Elements and Refined Abelian Stark Conjectures

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.