Prime-order complements in planar graph Jacobians

Nifa, Anass

doi:10.5281/zenodo.22865118

Preprint · v1

Prime-order complements in planar graph Jacobians

Anass Nifa5 August 2026

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Abstract

Let $(A,b)$ be a finite abelian group with a nondegenerate symmetric pairing into $\mathbb{Q}/\mathbb{Z}$ , and let $r(A)$ be its least number of generators. For infinitely many primes $\ell$ , we realize $(A,b)\perp\langle\kappa/\ell\rangle$ as the entire paired Jacobian of a simple planar biconnected graph, with one nonzero integer $\kappa$ independent of $\ell$ . The primes avoid any prescribed finite set. Every vertex has degree two or three. The graphs have arbitrarily large prescribed girth, treewidth at most two, and cycle rank $\max\{1,r(A)\}$ , which is optimal. For each cycle rank they are subdivisions of one fixed planar multigraph. We construct a family whose complementary orders form a primitive arithmetic progression; an explicit cofactor determines the complementary pairing. Dirichlet’s theorem gives the prime-order conclusion. A separate construction retains outerplanarity, allowing parallel edges and unbounded degree. Finally, for $r\ge3$ , the paired Jacobian of $K_{2^r}$ is the orthogonal sum of $2^{r-1}-1$ hyperbolic planes over $\mathbb{Z}/2^r\mathbb{Z}$ . The case $K_8$ contradicts the exceptional-pairing conjecture of Gaudet, Jensen, Ranganathan, Wawrykow and Weisman.

MSC 2020

05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
05C10: Planar graphs; geometric and topological aspects of graph theory
11E08: Quadratic forms over local rings and fields
11N13: Primes in congruence classes
20K01: Finite abelian groups

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22865118
All versions: 10.5281/zenodo.22865117
Subjects: Graph Jacobian; monodromy pairing; hyperbolic form; planar graph; prime-order completion; treewidth; finite abelian group
Language: English

Cite this paper

Anass Nifa. Prime-order complements in planar graph Jacobians. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22865118

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References

As listed in the manuscript bibliography.

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arXiv v2 · Source · DOI

[reference-1] [1]
R. Bacher, P. de la Harpe and T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. France 125 (1997), no. 2, 167–198. 10.24033/bsmf.2303.
DOI

[reference-2] [2]
S. Bosch and D. Lorenzini, Grothendieck's pairing on component groups of Jacobians, Invent. Math. 148 (2002), no. 2, 353–396. 10.1007/s002220100195.
DOI

[reference-3] [3]
H. Chen and B. Mohar, The sandpile group of a polygon flower, Discrete Appl. Math. 270 (2019), 68–82. 10.1016/j.dam.2019.07.020. arXiv:1907.08450v1 (2019).
arXiv v1 · DOI

[reference-4] [4]
H. Chen and B. Mohar, The sandpile group of polygon rings and twisted polygon rings, Graphs Combin. 38 (2022), no. 4, Paper No. 113. 10.1007/s00373-022-02514-x. arXiv:2011.08702v1 (2020).
arXiv v1 · DOI

[reference-5] [5]
J. Clancy, N. Kaplan, T. Leake, S. Payne and M. M. Wood, On a Cohen–Lenstra heuristic for Jacobians of random graphs, J. Algebraic Combin. 42 (2015), no. 3, 701–723. 10.1007/s10801-015-0598-x. arXiv:1402.5129v2.
arXiv v2 · DOI

[reference-6] [6]
P. G. Lejeune Dirichlet, There are infinitely many prime numbers in all arithmetic progressions with first term and difference coprime, English translation by R. Stephan of the 1837 paper, arXiv:0808.1408v2 (2014).
arXiv v2

[reference-7] [7]
L. Gaudet, D. Jensen, D. Ranganathan, N. Wawrykow and T. Weisman, Realization of groups with pairing as Jacobians of finite graphs, Ann. Comb. 22 (2018), no. 4, 781–801. 10.1007/s00026-018-0406-0. The conjecture and theorem numbering used here is that of arXiv:1410.5144v2 (18 September 2017).
arXiv v2 · DOI

[reference-8] [8]
J. E. Greene, The lens space realization problem, Ann. of Math. (2) 177 (2013), no. 2, 449–511. 10.4007/annals.2013.177.2.3.
DOI

[reference-9] [9]
E. Hodges, The distribution of sandpile groups of random graphs with their pairings, Trans. Amer. Math. Soc. 377 (2024), no. 12, 8769–8815. 10.1090/tran/9244. The numbering cited here is that of arXiv:2311.07078v1 (2023).
arXiv v1 · DOI

[reference-10] [10]
R. Miranda, Nondegenerate symmetric bilinear forms on finite abelian 2-groups, Trans. Amer. Math. Soc. 284 (1984), no. 2, 535–542. 10.1090/S0002-9947-1984-0743731-1.
DOI

[reference-11] [11]
H. H. Nguyen and M. M. Wood, Local and global universality of random matrix cokernels, Math. Ann. 391 (2025), 5117–5210. 10.1007/s00208-024-03050-0. The numbering cited here is that of arXiv:2210.08526v1 (2022).
arXiv v1 · DOI

[reference-12] [12]
A. Nifa, Homocyclic Jacobians of planar biconnected graphs, preprint, 13 pp., 10.5281/zenodo.22859833.
DOI

[reference-13] [13]
J. Shen, Quantative universality for cokernels of matrices with symmetries, preprint, arXiv:2601.09704v1 (2026), Theorem 1.2.
arXiv v1

[reference-14] [14]
F. Shokrieh, The monodromy pairing and discrete logarithm on the Jacobian of finite graphs, J. Math. Cryptol. 4 (2010), no. 1, 43–56. 10.1515/JMC.2010.002.
DOI

[reference-15] [15]
C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), no. 4, 281–298. 10.1016/0040-9383(63)90012-0.
DOI

[reference-16] [16]
M. M. Wood, The distribution of sandpile groups of random graphs, J. Amer. Math. Soc. 30 (2017), no. 4, 915–958. 10.1090/jams/866. arXiv:1402.5149v2. See also the author's errata.
arXiv v2 · Source · DOI