VOL. 86 | 2020 On Stark elements of arbitrary weight and their $p$-adic families
David Burns, Masato Kurihara, Takamichi Sano

Editor(s) Masato Kurihara, Kenichi Bannai, Tadashi Ochiai, Takeshi Tsuji

Adv. Stud. Pure Math., 2020: 113-140 (2020) DOI: 10.2969/aspm/08610113


In this article we shall generalize the theory of Stark elements to arbitrary ‘weight’. This means that we define canonical generalized Stark elements that correspond to the leading terms at arbitrary integers of $L$-series attached to the multiplicative group. We investigate their basic properties, and formulate both a conjecture on the precise integrality condition that these Stark elements satisfy and a conjecture that describes the Galois module structures of certain cohomology groups as a generalization of the refined Rubin-Stark conjecture we formulated in an earlier article ‘On zeta elements for $\mathbb{G}_m$’ in Doc. Math. 21 (2016). We then prove these conjectures in several interesting cases. The integrality condition studied here will then play an important role in a subsequent article of the first and third authors which formulates a precise conjectural congruence relation between the generalized Stark elements of differing weights that are defined here.


Published: 1 January 2020
First available in Project Euclid: 12 January 2021

Digital Object Identifier: 10.2969/aspm/08610113

Primary: 11S40
Secondary: 11R29 , 11R34 , 11R70

Keywords: generalized Stark elements , Stark's Conjecture

Rights: Copyright © 2020 Mathematical Society of Japan


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