Natural spectra and arithmetic of lens spaces

Nifa, Anass

doi:10.5281/zenodo.23313777

Preprint · v1

Natural spectra and arithmetic of lens spaces

Anass Nifa2 September 2026

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Abstract

The lens spaces $L(49;1,6,15)$ and $L(49;1,6,20)$ have identical natural spectra precisely for five irreducible $\mathrm{SO}(5)$ -types. This proves the finiteness conjecture of Lauret, Miatello and Rossetti. The types admitting bounded or compact self-adjoint corrections are classified, and the least operator norm and essential norm are determined for every one-row type on the bounded-correction locus. The distinction is arithmetic: exact equality requires equality of each Casimir eigenvalue multiplicity, whereas bounded correction requires equality of the total multiplicities in each separated quadratic cluster. Periodic character identities give both conditions and the asymptotic number of unmatched frequencies. For $L(p^2;1,1\pm p,1\pm3p)$ , the bounded-correction classification holds for every prime $p\ge5$ . For flat bundles on $L(q;1^m,a)$ , with $q\ge3$ odd, $m\ge1$ , and $a$ a unit modulo $q$ , equality of connection spectra is characterized in terms of the holonomies. For effective prime-square lens orbifolds, the scalar spectrum determines the gcd classes of the rotation weights and implies isometry of the corresponding proper covers at orders $9$ and $25$ .

MSC 2020

58J53Primary: Isospectrality
22E46Secondary: Semisimple Lie groups and their representations
11F72Secondary: Spectral theory; Selberg trace formula
11H06Secondary: Lattices and convex bodies

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.23313777
Subjects: Lens spaces; natural spectra; spectral rigidity; Casimir operators; congruence lattices; lens orbifolds
Language: English

Cite this paper

Anass Nifa. Natural spectra and arithmetic of lens spaces. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.23313777

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References

As listed in the manuscript bibliography.

[1]
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DOI
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DOI
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DOI · arXiv v3
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DOI · arXiv
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DOI · arXiv
[13]
E. A. Lauret and R. J. Miatello, Strong multiplicity one theorems for locally homogeneous spaces of compact type, Proceedings of the American Mathematical Society 148 (2020), 3163–3173. doi:10.1090/proc/14980. arXiv:1804.08288.
DOI · arXiv
[14]
E. A. Lauret and R. J. Miatello, Strong representation equivalence for compact symmetric spaces of real rank one, Pacific Journal of Mathematics 314 (2021), no. 2, 333–373. doi:10.2140/pjm.2021.314.333. arXiv:1812.09606.
DOI · arXiv
[15]
E. A. Lauret, R. J. Miatello and J. P. Rossetti, Spectra of lens spaces from 1-norm spectra of congruence lattices, International Mathematics Research Notices 2016 (2016), no. 4, 1054–1089. doi:10.1093/imrn/rnv159. arXiv:1311.7167.
DOI · arXiv
[16]
E. A. Lauret, R. J. Miatello and J. P. Rossetti, Recent results on the spectra of lens spaces, São Paulo Journal of Mathematical Sciences 15 (2021), 240–267. doi:10.1007/s40863-019-00154-3. arXiv:1904.01146v2.
DOI · arXiv v2
[17]
E. A. Lauret and F. Rossi Bertone, Weight multiplicity formulas for bivariate representations of classical Lie algebras, Journal of Mathematical Physics 59 (2018), no. 8, 081705. doi:10.1063/1.5043305. arXiv:1708.07905.
DOI · arXiv
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G. Teschl, Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators, second edition, Graduate Studies in Mathematics 157, American Mathematical Society, Providence, RI, 2014.

[reference-1] [1]
A. Álzaga and E. A. Lauret, Isospectral spherical space forms and orbifolds of highest volume, Revista de la Unión Matemática Argentina 69 (2026), no. 1, 71–91. doi:10.33044/revuma.4943.
DOI

[reference-2] [2]
E. A. Azoff and C. Davis, On distances between unitary orbits of self-adjoint operators, Acta Scientiarum Mathematicarum (Szeged) 47 (1984), 419–439.

[reference-3] [3]
N. S. Bari and E. Hunsicker, Isospectrality for orbifold lens spaces, Canadian Journal of Mathematics 72 (2020), no. 2, 281–325. doi:10.4153/S0008414X19000178.
DOI

[reference-4] [4]
T. Bogart, J. Goodrick and K. Woods, Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior, Discrete Analysis 2017 (2017), Paper No. 4, 34 pp. doi:10.19086/da.1254.
DOI

[reference-5] [5]
D. DeFord and P. G. Doyle, Cyclic groups with the same Hodge series, Revista de la Unión Matemática Argentina 59 (2018), no. 2, 241–254.

[reference-6] [6]
A. Dolzmann and T. Sturm, Parametric systems of linear congruences, in: Computer Algebra in Scientific Computing CASC 2001, Springer, Berlin, Heidelberg, 2001, 149–166. doi:10.1007/978-3-642-56666-0_12.
DOI

[reference-7] [7]
P. Etingof, 18.745: Lie Groups and Lie Algebras, I, lecture notes, Massachusetts Institute of Technology. https://math.mit.edu/~etingof/lnlg.pdf.

[reference-8] [8]
H. Kamiya, A. Takemura and H. Terao, Periodicity of hyperplane arrangements with integral coefficients modulo positive integers, Journal of Algebraic Combinatorics 27 (2008), 317–330. doi:10.1007/s10801-007-0091-2.
DOI

[reference-9] [9]
H. Kamiya, A. Takemura and H. Terao, Periodicity of non-central integral arrangements modulo positive integers, Annals of Combinatorics 15 (2011), 449–464. doi:10.1007/s00026-011-0105-6. arXiv:0803.2755.
DOI · arXiv

[reference-10] [10]
E. A. Lauret, Spectra of orbifolds with cyclic fundamental groups, Annals of Global Analysis and Geometry 50 (2016), 1–28. doi:10.1007/s10455-016-9498-0. arXiv:1510.05948v3.
DOI · arXiv v3

[reference-11] [11]
E. A. Lauret, The spectrum on p-forms of a lens space, Geometriae Dedicata 197 (2018), 107–122. doi:10.1007/s10711-018-0322-9. arXiv:1604.02471.
DOI · arXiv

[reference-12] [12]
E. A. Lauret, A computational study on lens spaces isospectral on forms, Experimental Mathematics 30 (2021), no. 2, 268–282. doi:10.1080/10586458.2018.1538908. arXiv:1703.03077.
DOI · arXiv

[reference-13] [13]
E. A. Lauret and R. J. Miatello, Strong multiplicity one theorems for locally homogeneous spaces of compact type, Proceedings of the American Mathematical Society 148 (2020), 3163–3173. doi:10.1090/proc/14980. arXiv:1804.08288.
DOI · arXiv

[reference-14] [14]
E. A. Lauret and R. J. Miatello, Strong representation equivalence for compact symmetric spaces of real rank one, Pacific Journal of Mathematics 314 (2021), no. 2, 333–373. doi:10.2140/pjm.2021.314.333. arXiv:1812.09606.
DOI · arXiv

[reference-15] [15]
E. A. Lauret, R. J. Miatello and J. P. Rossetti, Spectra of lens spaces from 1-norm spectra of congruence lattices, International Mathematics Research Notices 2016 (2016), no. 4, 1054–1089. doi:10.1093/imrn/rnv159. arXiv:1311.7167.
DOI · arXiv

[reference-16] [16]
E. A. Lauret, R. J. Miatello and J. P. Rossetti, Recent results on the spectra of lens spaces, São Paulo Journal of Mathematical Sciences 15 (2021), 240–267. doi:10.1007/s40863-019-00154-3. arXiv:1904.01146v2.
DOI · arXiv v2

[reference-17] [17]
E. A. Lauret and F. Rossi Bertone, Weight multiplicity formulas for bivariate representations of classical Lie algebras, Journal of Mathematical Physics 59 (2018), no. 8, 081705. doi:10.1063/1.5043305. arXiv:1708.07905.
DOI · arXiv

[reference-18] [18]
G. Teschl, Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators, second edition, Graduate Studies in Mathematics 157, American Mathematical Society, Providence, RI, 2014.