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.
Revised:
Accepted:
Published online:
Mots-clés : graphs, Jacobian groups, Ihara zeta functions, Fitting ideals
Kataoka, Takenori 1

@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 -
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] Fitting ideals of class groups for CM abelian extensions, Algebra Number Theory, Volume 17 (2023) no. 11, pp. 1901-1924 | DOI | MR | Zbl
[2] Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. Math., Volume 215 (2007) no. 2, pp. 766-788 | DOI | MR | Zbl
[3] 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] Divisors and sandpiles: An introduction to chip-firing, American Mathematical Society, Providence, RI, 2018, xiv+325 pages | DOI | MR
[5] Iwasawa Theory of Jacobians of Graphs, Algebr. Comb., Volume 5 (2022) no. 5, pp. 827-848 | Numdam | MR | Zbl
[6] 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] Tate sequences and Fitting ideals of Iwasawa modules, Algebra i Analiz, Volume 27 (2015) no. 6, pp. 117-149 | MR
[8] The second syzygy of the trivial -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] The special value of Artin-Ihara -functions, Proc. Amer. Math. Soc., Volume 152 (2024) no. 2, pp. 501-514 | DOI | MR | Zbl
[10] On -extensions of algebraic number fields, Bull. Amer. Math. Soc., Volume 65 (1959), pp. 183-226 | DOI | MR | Zbl
[11] 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] -extensions of CM-fields and cyclotomic invariants, J. Number Theory, Volume 12 (1980) no. 4, pp. 519-528 | DOI | MR | Zbl
[13] On the growth of the Jacobians in -voltage covers of graphs, 2022 | arXiv
[14] On abelian -towers of multigraphs III, Ann. Math. Qué., Volume 48 (2024) no. 1, pp. 1-19 | DOI | MR
[15] Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 323, Springer-Verlag, Berlin, 2008, xvi+825 pages | DOI | MR
[16] Finite free resolutions, Cambridge University Press, Cambridge-New York-Melbourne, 1976, xii+271 pages (Cambridge Tracts in Mathematics, No. 71) | DOI | MR
[17] An analogue of Kida’s formula in graph theory, 2022 | arXiv
[18] Trees, Springer-Verlag, Berlin-New York, 1980, ix+142 pages (translated from the French by John Stillwell) | DOI | MR
[19] 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] Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1997, xiv+487 pages | DOI | MR
Cited by Sources: