Browse issues
no. 3
Fitting ideals of Jacobian groups of graphs
Kataoka, Takenori1
1 Tokyo University of Science Department of Mathematics, Faculty of Science Division II 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601 (Japan)
Algebraic Combinatorics, Volume 7 (2024) no. 3, pp. 597-625.

The Jacobian group of a graph is a finite abelian group through which we can study the graph in an algebraic way. When the graph is a finite abelian covering of another graph, the Jacobian group is equipped with the action of the Galois group. In this paper, we study the Fitting ideal of the Jacobian group as a module over the group ring. We also study the corresponding question for infinite coverings. Additionally, this paper includes module-theoretic approach to Iwasawa theory for graphs.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.355
Classification: 05C25, 11R23, 16E05
Keywords: graphs, Jacobian groups, Ihara zeta functions, Fitting ideals

Kataoka, Takenori 1

1 Tokyo University of Science Department of Mathematics, Faculty of Science Division II 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601 (Japan)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2024__7_3_597_0,
     author = {Kataoka, Takenori},
     title = {Fitting ideals of {Jacobian} groups of graphs},
     journal = {Algebraic Combinatorics},
     pages = {597--625},
     publisher = {The Combinatorics Consortium},
     volume = {7},
     number = {3},
     year = {2024},
     doi = {10.5802/alco.355},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.355/}
}
TY  - JOUR
AU  - Kataoka, Takenori
TI  - Fitting ideals of Jacobian groups of graphs
JO  - Algebraic Combinatorics
PY  - 2024
SP  - 597
EP  - 625
VL  - 7
IS  - 3
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.355/
DO  - 10.5802/alco.355
LA  - en
ID  - ALCO_2024__7_3_597_0
ER  - 
%0 Journal Article
%A Kataoka, Takenori
%T Fitting ideals of Jacobian groups of graphs
%J Algebraic Combinatorics
%D 2024
%P 597-625
%V 7
%N 3
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.355/
%R 10.5802/alco.355
%G en
%F ALCO_2024__7_3_597_0
Kataoka, Takenori. Fitting ideals of Jacobian groups of graphs. Algebraic Combinatorics, Volume 7 (2024) no. 3, pp. 597-625. doi : 10.5802/alco.355. https://alco.centre-mersenne.org/articles/10.5802/alco.355/

[1] Atsuta, Mahiro; Kataoka, Takenori Fitting ideals of class groups for CM abelian extensions, Algebra Number Theory, Volume 17 (2023) no. 11, pp. 1901-1924 | DOI | MR | Zbl

[2] Baker, Matthew; Norine, Serguei Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. Math., Volume 215 (2007) no. 2, pp. 766-788 | DOI | MR | Zbl

[3] Bourbaki, Nicolas Elements of mathematics. Algebra, Part I: Chapters 1-3, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1974, xxiii+709 pages (translated from the French) | MR

[4] Corry, Scott; Perkinson, David Divisors and sandpiles: An introduction to chip-firing, American Mathematical Society, Providence, RI, 2018, xiv+325 pages | DOI | MR

[5] Gonet, Sophia R. Iwasawa Theory of Jacobians of Graphs, Algebr. Comb., Volume 5 (2022) no. 5, pp. 827-848 | Numdam | MR | Zbl

[6] Greenberg, Ralph Iwasawa theory for elliptic curves, Arithmetic theory of elliptic curves (Cetraro, 1997) (Lecture Notes in Math.), Volume 1716, Springer, Berlin, 1999, pp. 51-144 | DOI | MR | Zbl

[7] Greither, C.; Kurihara, M. Tate sequences and Fitting ideals of Iwasawa modules, Algebra i Analiz, Volume 27 (2015) no. 6, pp. 117-149 | MR

[8] Greither, Cornelius; Kurihara, Masato; Tokio, Hibiki The second syzygy of the trivial G-module, and an equivariant main conjecture, Development of Iwasawa theory—the centennial of K. Iwasawa’s birth (Adv. Stud. Pure Math.), Volume 86, Math. Soc. Japan, Tokyo, 2020, pp. 317-349 | DOI | MR | Zbl

[9] Hammer, Kyle; Mattman, Thomas; Sands, Jonathan; Vallières, Daniel The special value u=1 of Artin-Ihara L-functions, Proc. Amer. Math. Soc., Volume 152 (2024) no. 2, pp. 501-514 | DOI | MR | Zbl

[10] Iwasawa, Kenkichi On Γ-extensions of algebraic number fields, Bull. Amer. Math. Soc., Volume 65 (1959), pp. 183-226 | DOI | MR | Zbl

[11] Kataoka, Takenori Fitting invariants in equivariant Iwasawa theory, Development of Iwasawa theory—the centennial of K. Iwasawa’s birth (Adv. Stud. Pure Math.), Volume 86, Math. Soc. Japan, Tokyo, 2020, pp. 413-465 | DOI | MR | Zbl

[12] Kida, Yûji l-extensions of CM-fields and cyclotomic invariants, J. Number Theory, Volume 12 (1980) no. 4, pp. 519-528 | DOI | MR | Zbl

[13] Kleine, Sören; Müller, Katharina On the growth of the Jacobians in ℤ p l -voltage covers of graphs, 2022 | arXiv

[14] McGown, Kevin; Vallières, Daniel On abelian ℓ-towers of multigraphs III, Ann. Math. Qué., Volume 48 (2024) no. 1, pp. 1-19 | DOI | MR

[15] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 323, Springer-Verlag, Berlin, 2008, xvi+825 pages | DOI | MR

[16] Northcott, D. G. Finite free resolutions, Cambridge University Press, Cambridge-New York-Melbourne, 1976, xii+271 pages (Cambridge Tracts in Mathematics, No. 71) | DOI | MR

[17] Ray, Anwesh; Vallières, Daniel An analogue of Kida’s formula in graph theory, 2022 | arXiv

[18] Serre, Jean-Pierre Trees, Springer-Verlag, Berlin-New York, 1980, ix+142 pages (translated from the French by John Stillwell) | DOI | MR

[19] Terras, Audrey Zeta functions of graphs: A stroll through the garden, Cambridge Studies in Advanced Mathematics, 128, Cambridge University Press, Cambridge, 2011, xii+239 pages | MR

[20] Washington, Lawrence C. Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1997, xiv+487 pages | DOI | MR

Cited by Sources: