Realization and rigidity of finite discriminant pairings

Nifa, Anass

doi:10.5281/zenodo.23224615

Preprint · v2

Realization and rigidity of finite discriminant pairings

Anass Nifa6 September 2026

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Abstract

Let $\mu(A,b)$ be the least cycle rank of a connected loopless multigraph whose Jacobian has a prescribed nonsingular symmetric pairing $(A,b)$ . For odd-order groups, $d(A)\le\mu(A,b)\le d(A)+2$ , where $d(A)$ is the minimum number of generators. Exact formulas hold for odd-primary pairings and for mixed-prime pairings with at most two elementary-divisor exponents at each prime. For every finite pairing, adjoining one cyclic pairing of prime order permits planar realization in cycle rank $\max\{1,d(A)\}$ . In rank at least two, the graphs subdivide a fixed ladder. The local pairings remain fixed while one path length varies through a primitive arithmetic progression of determinants. Prescribed prime residue and pairing classes are characterized by local determinant and Hasse invariants. Symmetric ladders give hyperelliptic Jacobians with prescribed positive monodromy pairings, up to one prime summand, at finitely many places of odd residue characteristic and in the least possible semistable dimension when $\max_vd(A_v)\ge2$ . The local fields remain unchanged. Finally, a rational homology three-sphere of first homology $\mathbb{Z}/2$ cannot embed locally flatly in a simply connected topological four-manifold with intersection lattice an orthogonal power of the positive Leech lattice.

MSC 2020

11E08: Quadratic forms over local rings and fields
11E12: Quadratic forms over global rings and fields
05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
14H40: Jacobians, Prym varieties
11G20: Curves over finite and local fields
57N35: Embeddings and immersions in topological manifolds

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.23224615
All versions: 10.5281/zenodo.22865117
Subjects: Discriminant pairing; integral lattice; graph Jacobian; monodromy; prime complement; locally flat embedding
Language: English

Cite this paper

Anass Nifa. Realization and rigidity of finite discriminant pairings. Preprint, Zenodo, v2 (2026). doi:10.5281/zenodo.23224615

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References

As listed in the manuscript bibliography.

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[reference-1] [1]
O. Amini, M. Baker, E. Brugallé and J. Rabinoff, Lifting harmonic morphisms II: tropical curves and metrized complexes, Algebra Number Theory 9 (2015), no. 2, 267–315. https://doi.org/10.2140/ant.2015.9.267.

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P. Aceto, D. McCoy and J. Park, Definite fillings of lens spaces, J. Differential Geom. 131 (2025), no. 1, 1–41. 10.4310/jdg/1755541518. https://arxiv.org/abs/2208.02586v2.
DOI

[reference-3] [3]
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), 167–198. https://doi.org/10.24033/bsmf.2303.

[reference-4] [4]
M. Baker and S. Norine, Harmonic morphisms and hyperelliptic graphs, Int. Math. Res. Not. IMRN 2009 (2009), no. 15, 2914–2955. 10.1093/imrn/rnp037.
DOI

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DOI

[reference-6] [6]
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DOI

[reference-7] [7]
H. Chen and B. Mohar, The sandpile group of polygon rings and twisted polygon rings, Graphs Combin. 38 (2022), no. 4, article 113. 10.1007/s00373-022-02514-x.
DOI

[reference-8] [8]
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DOI

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A. Landesman, A. A. Swaminathan, J. Tao and Y. Xu, Surjectivity of Galois representations in rational families of abelian varieties, Algebra Number Theory 13 (2019), 995–1038. https://doi.org/10.2140/ant.2019.13.995.

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arXiv v1 · DOI

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DOI

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arXiv v1

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I. Shimada, Involutions in the affine Conway group, arXiv:2608.29065v1, 29 August 2026. https://arxiv.org/abs/2608.29065v1.

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DOI

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DOI

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DOI

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arXiv

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X.-J. Zhu, Finite quadratic modules and lattices, Comm. Algebra 52 (2024), 604–629. https://doi.org/10.1080/00927872.2023.2245924. Preprint: https://arxiv.org/abs/2110.06783.