Shintani zeta functions and a refinement of Gross's leading term conjecture

TL;DR

This paper introduces Shintani data to axiomatize algebraic aspects of Shintani zeta functions, providing new insights into Gross's conjecture and proposing a refined conjecture involving Rubin-Stark elements.

Contribution

It develops the theory of Shintani data, proves the order of vanishing part of Gross's conjecture via this framework, and proposes a refined conjecture for the leading term involving higher rank Rubin-Stark elements.

Findings

Alternative proof of Gross's conjecture order of vanishing

Construction of a Shintani datum for Gross's conjecture

Proposal of a refined conjecture involving Rubin-Stark elements

Abstract

We introduce the notion of Shintani data, which axiomatizes algebraic aspects of Shintani zeta functions. We develop the general theory of Shintani data, and show that the order of vanishing part of Gross's conjecture follows from the existence of a Shintani datum. Recently, Dasgupta and Spiess proved the order of vanishing part of Gross's conjecture under certain conditions. We give an alternative proof of their result by constructing a certain Shintani datum. We also propose a refinement of Gross's leading term conjecture by using the theory of Shintani data. Out conjecture gives a conjectural construction of localized Rubin-Stark elements which can be regarded as a higher rank generalization of the conjectural construction of Gross-Stark units due to Dasgupta and Dasgupta-Spiess.