Prime-order complements in planar graph Jacobians
Abstract
Let (A,b) be a finite abelian group with a nondegenerate symmetric pairing into Q/Z, and let r(A) be its least number of generators. For infinitely many primes ℓ, we realize (A,b) ⊥ ⟨κ/ℓ⟩ as the entire paired Jacobian of a simple planar biconnected graph, with one nonzero integer κ independent of ℓ. 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 ≥ 3, the paired Jacobian of K_(2^r) is the orthogonal sum of 2^(r−1)−1 hyperbolic planes over Z/2^r Z. The case K_8 contradicts the exceptional-pairing conjecture of Gaudet, Jensen, Ranganathan, Wawrykow and Weisman. Preprint. MSC 2020: 05C25, 05C50, 05C10, 11E08, 11N13, 20K01.
Related
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BibTeX
@misc{nifa2026primeorder,
title = {{Prime-order complements in planar graph Jacobians}},
author = {Anass Nifa},
howpublished = {HAL},
year = {2026},
month = aug,
doi = {10.5281/zenodo.22865118},
url = {https://hal.science/hal-05769897v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Prime-order complements in planar graph Jacobians
AU - Anass Nifa
PY - 2026
DA - 2026/08/05
PB - HAL
DO - 10.5281/zenodo.22865118
UR - https://hal.science/hal-05769897v1
AB - Let (A,b) be a finite abelian group with a nondegenerate symmetric pairing into Q/Z, and let r(A) be its least number of generators. For infinitely many primes ℓ, we realize (A,b) ⊥ ⟨κ/ℓ⟩ as the entire paired Jacobian of a simple planar biconnected graph, with one nonzero integer κ independent of ℓ. 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 ≥ 3, the paired Jacobian of K_(2^r) is the orthogonal sum of 2^(r−1)−1 hyperbolic planes over Z/2^r Z. The case K_8 contradicts the exceptional-pairing conjecture of Gaudet, Jensen, Ranganathan, Wawrykow and Weisman. Preprint. MSC 2020: 05C25, 05C50, 05C10, 11E08, 11N13, 20K01.
ER -