A Chip-Firing and a Riemann-Roch Theorem on an Ultrametric Space

Atsushi Atsuji, Hiroshi Kaneko

Research output: Chapter in Book/Report/Conference proceedingConference contribution


A Riemann-Roch theorem on an edge-weighted infinite graph with local finiteness was established by the present authors in [1], where the spectral gap of Laplacian associated determined by the edge-weight was investigated as the corner stone of the proof. On the other hand, as for non-archimedean metric space, the Laplacians in the construction of Hunt processes such as in [3, 5] based on the Dirichlet space theory can be highlighted. However, in those studies, a positive edge-weight was given substantially between each pair of balls with an identical diameter with respect to the ultrametric and the spectral gap is infeasible. In the present article, we rethink the notion of chip-firing and show an upper bound of function given by accumulation of chip-firing to materialize a counterpart of the dimension of linear system in ultrametric space. In the final section of this article, we establish a Riemann-Roch theorem on an ultrametric space.

Original languageEnglish
Title of host publicationDirichlet Forms and Related Topics - In Honor of Masatoshi Fukushima’s Beiju, IWDFRT 2022
EditorsZhen-Qing Chen, Masayoshi Takeda, Toshihiro Uemura
Number of pages21
ISBN (Print)9789811946714
Publication statusPublished - 2022
EventInternational Conference on Dirichlet Forms and Related Topics, IWDFRT 2022 - Osaka, Japan
Duration: 2022 Aug 222022 Aug 26

Publication series

NameSpringer Proceedings in Mathematics and Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017


ConferenceInternational Conference on Dirichlet Forms and Related Topics, IWDFRT 2022


  • Dirichlet space
  • Laplacian
  • Riemann-Roch theorem
  • Ultrametric space
  • Weighted graph

ASJC Scopus subject areas

  • General Mathematics


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