On the growth of the Jacobians in p l -voltage covers of graphs
Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 1011-1038.

We investigate the growth of the p-part of the Jacobians in voltage covers of finite connected graphs, where the voltage group is isomorphic to p l for some l2, and we study analogues of a conjecture of Greenberg on the growth of class numbers in multiple p -extensions of number fields. Moreover we prove an Iwasawa main conjecture in this setting, and we study the variation of (generalised) Iwasawa invariants as one runs over the p l -covers of a fixed finite graph X. We discuss many examples; in particular, we construct examples with non-trivial Iwasawa invariants.

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DOI: 10.5802/alco.366
Classification: 11C30, 11R23, 05C25, 05C40, 05C50, 11C20
Keywords: Voltage cover of a graph, Greenberg’s conjecture, Iwasawa main conjecture, (generalised) Iwasawa invariants

Kleine, Sören 1; Müller, Katharina 2

1 Institut für Anwendungssicherheit Universität der Bundeswehr München Werner-Heisenberg-Weg 39 85577 Neubiberg Germany
2 Institut für Theoretische Informatik Mathematik und Operations Research Universität der Bundeswehr München Werner-Heisenberg-Weg 39 85577 Neubiberg Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Kleine, Sören; Müller, Katharina. On the growth of the Jacobians in $\mathbb{Z}_p^l$-voltage covers of graphs. Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 1011-1038. doi : 10.5802/alco.366. https://alco.centre-mersenne.org/articles/10.5802/alco.366/

[1] Babaĭcev, Vladimir A. On some questions in the theory of Γ-extensions of algebraic number fields. II., Math. USSR, Izv., Volume 16 (1976), pp. 675-685 | DOI | Zbl

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

[3] 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

[4] DuBose, Sage; Vallières, Daniel On d -towers of graphs, Algebr. Comb., Volume 6 (2023) no. 5, pp. 1331-1346 | DOI | MR | Zbl

[5] Fukuda, Takashi Remarks on Z p -extensions of number fields, Proc. Japan Acad. Ser. A Math. Sci., Volume 70 (1994) no. 8, pp. 264-266 http://projecteuclid.org/euclid.pja/1195510924 | MR | Zbl

[6] Gonet, Sophia R. Jacobians of Finite and Infinite Voltage Covers of Graphs, Ph. D. Thesis, The University of Vermont and State Agricultural College (2021) (266 pages)

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

[8] Greenberg, Ralph The Iwasawa invariants of Γ-extensions of a fixed number field., Am. J. Math., Volume 95 (1973), pp. 204-214 | DOI | Zbl

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

[10] Iwasawa, Kenkichi On the μ-invariants of Z -extensions, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 1-11 | MR | Zbl

[11] Kleine, Sören Local behavior of Iwasawa’s invariants., Int. J. Number Theory, Volume 13 (2017) no. 4, pp. 1013-1036 | DOI | Zbl

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

[13] Kleine, Sören; Matar, Ahmed Boundedness of Iwasawa invariants of fine Selmer groups and Selmer groups, Results Math., Volume 78 (2023) no. 4, Paper no. 148, 42 pages | DOI | MR | Zbl

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

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

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

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

[18] 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

[19] Ouellette, Diane Valérie Schur complements and statistics, Linear Algebra Appl., Volume 36 (1981), pp. 187-295 | DOI | MR | Zbl

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

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

[22] Wei, Yunlan; Jiang, Xiaoyu; Jiang, Zhaolin; Shon, Sugoog Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices, Adv. Difference Equ. (2019), Paper no. 410, 11 pages | DOI | MR | Zbl

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