2016/02/01 by Minoru Hirose, Hirose, Minoru
Mathematics · #11R42 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11R42
paper · pdf · doi:10.48550/arxiv.1602.00666
23 pages
arxiv created 2016/02/01 · openalex publication_date 2016/02/01 · arxiv updated 2016/02/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.