TY - GEN

T1 - A Chip-Firing and a Riemann-Roch Theorem on an Ultrametric Space

AU - Atsuji, Atsushi

AU - Kaneko, Hiroshi

N1 - Funding Information:
Acknowledgements This work was supported by JSPS Grant-in-Aid for Scientific Research (C) Grant Numbers JP21K03277 and JP21K03299.
Publisher Copyright:
© 2022, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Dirichlet space

KW - Laplacian

KW - Riemann-Roch theorem

KW - Ultrametric space

KW - Weighted graph

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U2 - 10.1007/978-981-19-4672-1_2

DO - 10.1007/978-981-19-4672-1_2

M3 - Conference contribution

AN - SCOPUS:85137981737

SN - 9789811946714

T3 - Springer Proceedings in Mathematics and Statistics

SP - 23

EP - 43

BT - Dirichlet Forms and Related Topics - In Honor of Masatoshi Fukushima’s Beiju, IWDFRT 2022

A2 - Chen, Zhen-Qing

A2 - Takeda, Masayoshi

A2 - Uemura, Toshihiro

PB - Springer

T2 - International Conference on Dirichlet Forms and Related Topics, IWDFRT 2022

Y2 - 22 August 2022 through 26 August 2022

ER -