Iwasawa Theory of Jacobians of Graphs

Gonet, Sophia R.

Algebraic Combinatorics, Tome 5 (2022) no. 5, pp. 827-848.

The Jacobian group (also known as the critical group or sandpile group) is an important invariant of a finite, connected graph $X$; it is a finite abelian group whose cardinality is equal to the number of spanning trees of $X$ (Kirchhoff’s Matrix Tree Theorem). A specific type of covering graph, called a derived graph, that is constructed from a voltage graph with voltage group $G$ is the object of interest in this paper.

Towers of derived graphs are studied by using aspects of classical Iwasawa Theory (from number theory). Formulas for the orders of the Sylow $p$-subgroups of Jacobians in an infinite voltage $p$-tower, for any prime $p$, are obtained in terms of classical $\mu$ and $\lambda$ invariants by using the decomposition of a finitely generated module over the Iwasawa Algebra.

Mots clés : Jacobian group, Iwasawa Theory, Laplacian, voltage graph, Picard group

[1] Bacher, Roland; de la Harpe, Pierre; Nagnibeda, Tatiana The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. France, Volume 125 (1995), pp. 167-198 | DOI | MR | Zbl

[2] Backman, Spencer; Bakek, Matthew; Yuen, Chi Ho Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory, Forum Math. Sigma, Volume 7 (2019), E45, 37 pages | MR | Zbl

[3] Bai, Hua On the critical group of the $n$-cube, Linear Algebra Appl., Volume 369 (2003), pp. 251-261 | DOI | MR | Zbl

[4] Biggs, Norman Chip-firing and the critical group of a graph, J. Algebraic Combin., Volume 9 (1999), pp. 25-45 | DOI | MR | Zbl

[5] Biggs, Norman The Critical group from a cryptographic perspective, Bull. Lond. Math. Soc., Volume 39 (2007), pp. 829-836 | DOI | MR | Zbl

[6] Chandler, David B.; Sin, Peter; Xiang, Qing The Smith and Critical Groups of Paley Graphs, J. Algebraic Combin., Volume 41 (2015), pp. 1013-1022 | DOI | MR | Zbl

[7] Chepuri, Sunita; Dowd, Christopher J.; Hardt, Andrew; Michel, Gregory; Zhang, Sylvester W.; Zhang, Valerie Arborescences of covering graphs, Algebr. Comb., Volume 5 (2022) no. 2, pp. 319-346 | DOI | MR | Zbl

[8] Christianson, Hans; Reiner, Victor The Critical Group of a Threshold Graph, Linear Algebra Appl., Volume 394 (2002), pp. 233-244 | DOI | MR | Zbl

[9] Coates, John; Sujatha, Ramadorai Cyclotomic fields and zeta values, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006, x+113 pages | MR

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

[11] Dhar, Deepak Self-organized critical state of sandpile automaton models, Phys. Rev. Lett., Volume 64 (1990), pp. 1613-1616 | DOI | MR | Zbl

[12] Ducey, Joshua; Hill, Ian; Sin, Peter The Critical Group of the Kneser Graph on $2$-subsets of an $n$-Element Set, Linear Algebra Appl., Volume 546 (2018), pp. 154-168 | DOI | MR | Zbl

[13] Ducey, Joshua E.; Jalil, Deelan M. Integer invariants of abelian Cayley graphs, Linear Algebra Appl., Volume 445 (2014), pp. 316-325 | DOI | MR | Zbl

[14] Dummit, David S.; Foote, Richard M. Abstract algebra, John Wiley & Sons, Inc., Hoboken, NJ, 2004, xii+932 pages | MR

[15] Forrás, Bence Iwasawa Theory, 2020 (http://bforras.eu/docs/Forrás_Iwasawa_Theory_blue.pdf)

[16] Gonet, Sophia Jacobians of Finite and Infinite Voltage Covers of Graphs, Ph. D. Thesis, University of Vermont (2021)

[17] Hammer, Kyle; Mattman, Thomas W.; Sands, Jonathan W.; Vallières, Daniel The special value $u=1$ of Artin-Ihara $\mathrm{L}$-functions, 2019 | arXiv

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

[19] Jacobson, Brian; Niedermaier, Andrew; Reiner, Victor Critical groups for complete multipartite graphs and Cartesian products of complete graphs, J. Graph Theory, Volume 44 (2003), pp. 231-250 | DOI | MR | Zbl

[20] Klivans, Caroline J. The mathematics of chip-firing, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2019, xii+295 pages | MR

[21] Kurokawa, Nobushige; Kurihara, Masato; Saito, Takeshi Number theory. 3. Iwasawa theory and modular forms, Translated from the Japanese by Masato Kuwata, Iwanami Series in Modern Mathematics, Translations of Mathematical Monographs, 242, American Mathematical Society, Providence, RI, 2012, xiv+226 pages | DOI

[22] Kwon, Young Soo; Mednykh, Aleksandr D.; Mednykh, Ilya A. On Jacobian group and complexity of the generalized Petersen graph GP(n,k) through Chebyshev polynomials, Linear Algebra Appl., Volume 529 (2017), pp. 355-373 | DOI | MR | Zbl

[23] McGown, Kevin J.; Vallières, Daniel On abelian $\ell$-towers of multigraphs II, 2021 | arXiv

[24] McGown, Kevin J.; Vallières, Daniel On abelian $\ell$-towers of multigraphs III, 2021 | arXiv

[25] Reiner, Victor; Tseng, Dennis Critical Groups of Coverings, Voltage and Signed Graphs, Discrete Math., Volume 318 (2014), pp. 10-40 | DOI | MR | Zbl

[26] Sharifi, Romyar Iwasawa Theory (https://www.math.ucla.edu/~sharifi/iwasawa.pdf)

[27] Terras, Audrey Zeta Functions of Graphs, 1st Ed., Cambridge University Press, 2011

[28] Vallières, Daniel On abelian $\ell$-towers of multigraphs, Ann. Math. Qué., Volume 45 (2021) no. 2, pp. 433-452 | DOI | MR | Zbl

[29] Washington, Lawrence C. Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1982, xi+389 pages | DOI