On $\mathbb{Z}_{\ell }^{d}$-towers of graphs

DuBose, Sage; Vallières, Daniel

doi:10.5802/alco.304

ℤ_{ℓ}^{d}

-towers of graphs

DuBose, Sage ¹; Vallières, Daniel ¹

¹ California State University Mathematics and Statistics Department Chico CA 95929 USA

Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1331-1346.

Abstract

Let $ℓ$ be a rational prime. We show that an analogue of a conjecture of Greenberg in graph theory holds true. More precisely, we show that when $n$ is sufficiently large, the $ℓ$ -adic valuation of the number of spanning trees at the $n$ th layer of a $ℤ_{ℓ}^{d}$ -tower of graphs is given by a polynomial in $ℓ^{n}$ and $n$ with rational coefficients of total degree at most $d$ and of degree in $n$ at most one.

Received: 2022-07-04
Revised: 2023-01-18
Accepted: 2023-02-15
Published online: 2023-11-07

DOI: 10.5802/alco.304

Classification: 05C25, 11R18, 11R23, 11Z05
Keywords: Ihara zeta functions, Iwasawa theory, spanning trees

Author's affiliations:

DuBose, Sage ¹; Vallières, Daniel ¹

¹ California State University Mathematics and Statistics Department Chico CA 95929 USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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DuBose, Sage; Vallières, Daniel. On $\mathbb{Z}_{\ell }^{d}$-towers of graphs. Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1331-1346. doi : 10.5802/alco.304. https://alco.centre-mersenne.org/articles/10.5802/alco.304/

References
Cited by

[1] Cuoco, Albert A.; Monsky, Paul Class numbers in $Z_{p}^{d}$ -extensions, Math. Ann., Volume 255 (1981) no. 2, pp. 235-258 | DOI | MR | Zbl

[2] Gold, R.; Kisilevsky, H. On geometric $Z_{p}$ -extensions of function fields, Manuscripta Math., Volume 62 (1988) no. 2, pp. 145-161 | DOI | MR | Zbl

[3] Gonet, Sophia R. Jacobians of Finite and Infinite Voltage Covers of Graphs. Thesis (Ph.D.)–The University of Vermont and State Agricultural College, ProQuest LLC, Ann Arbor, MI, 2021, 266 pages | MR

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

[5] Gross, Jonathan L.; Tucker, Thomas W. Generating all graph coverings by permutation voltage assignments, Discrete Math., Volume 18 (1977) no. 3, pp. 273-283 | DOI | MR | Zbl

[6] Gross, Jonathan L.; Tucker, Thomas W. Topological graph theory, Dover Publications, Inc., Mineola, NY, 2001, xvi+361 pages | MR

[7] Hashimoto, Ki-ichiro On zeta and $L$ -functions of finite graphs, Internat. J. Math., Volume 1 (1990) no. 4, pp. 381-396 | DOI | MR | Zbl

[8] Ihara, Yasutaka On discrete subgroups of the two by two projective linear group over $𝔭$ -adic fields, J. Math. Soc. Japan, Volume 18 (1966), pp. 219-235 | DOI | MR | Zbl

[9] Iwasawa, Kenkichi On $ℤ_{l}$ -extensions of algebraic number fields, Ann. of Math. (2), Volume 98 (1973), pp. 246-326 | DOI | MR | Zbl

[10] Kleine, Sören Generalised Iwasawa invariants and the growth of class numbers, Forum Math., Volume 33 (2021) no. 1, pp. 109-127 | DOI | MR | Zbl

[11] Lei, Antonio; Vallières, Daniel The non- $ℓ$ -part of the number of spanning trees in abelian $ℓ$ -towers of multigraphs, Res. Number Theory, Volume 9 (2023) no. 1, Paper no. 18, 16 pages | DOI | MR | Zbl

[12] McGown, Kevin; Vallières, Daniel On abelian $ℓ$ -towers of multigraphs II, Ann. Math. Qué., Volume 47 (2023) no. 2, pp. 461-473 | DOI | MR | Zbl

[13] McGown, Kevin J.; Vallières, Daniel On abelian $ℓ$ -towers of multigraphs III, To appear in Annales Mathématiques du Québec (2022)

[14] Monsky, Paul On $p$ -adic power series, Math. Ann., Volume 255 (1981) no. 2, pp. 217-227 | DOI | MR | Zbl

[15] Monsky, Paul Class numbers in $Z_{p}^{d}$ -extensions. II, Math. Z., Volume 191 (1986) no. 3, pp. 377-395 | DOI | MR

[16] Monsky, Paul Class numbers in $Z_{p}^{d}$ -extensions. III, Math. Z., Volume 193 (1986) no. 4, pp. 491-514 | DOI | MR | Zbl

[17] Monsky, Paul Class numbers in $Z_{p}^{d}$ -extensions. IV, Math. Z., Volume 196 (1987) no. 4, pp. 547-572 | DOI | MR | Zbl

[18] Monsky, Paul Fine estimates for the growth of $e_{n}$ in $Z_{p}^{d}$ -extensions, Algebraic number theory (Adv. Stud. Pure Math.), Volume 17, Academic Press, Boston, MA, 1989, pp. 309-330 | DOI | MR

[19] Neukirch, Jürgen Algebraic number theory. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder, Grundlehren der Mathematischen Wissenschaften, 322, Springer-Verlag, Berlin, 1999, xviii+571 pages | DOI | MR

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

[21] Northshield, Sam A note on the zeta function of a graph, J. Combin. Theory Ser. B, Volume 74 (1998) no. 2, pp. 408-410 | DOI | MR | Zbl

[22] Serre, Jean-Pierre Arbres, amalgames, ${SL}_{2}$ . Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass, Astérisque, 46, Société Mathématique de France, Paris, 1977, 189 pp. (1 plate) pages | MR

[23] Stark, H. M.; Terras, A. A. Zeta functions of finite graphs and coverings, Adv. Math., Volume 121 (1996) no. 1, pp. 124-165 | DOI | MR | Zbl

[24] Stark, H. M.; Terras, A. A. Zeta functions of finite graphs and coverings. II, Adv. Math., Volume 154 (2000) no. 1, pp. 132-195 | DOI | MR | Zbl

[25] Sunada, Toshikazu $L$ -functions in geometry and some applications, Curvature and topology of Riemannian manifolds (Katata, 1985) (Lecture Notes in Math.), Volume 1201, Springer, Berlin, 1986, pp. 266-284 | DOI | MR | Zbl

[26] Sunada, Toshikazu Topological crystallography.With a view towards discrete geometric analysis, Surveys and Tutorials in the Applied Mathematical Sciences, 6, Springer, Tokyo, 2013, xii+229 pages | DOI | MR

[27] 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

[28] The Sage Developers SageMath, the Sage Mathematics Software System (Version 8.5) (2018) (https://www.sagemath.org)

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

[30] Wan, Daqing Class numbers and $p$ -ranks in $ℤ_{p}^{d}$ -towers, J. Number Theory, Volume 203 (2019), pp. 139-154 | DOI | MR | Zbl

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