Joint distribution of the cokernels of random p-adic matrices

Jungin Lee

doi:10.1515/forum-2022-0209

Article

Joint distribution of the cokernels of random p-adic matrices

Jungin Lee

Published/Copyright: May 3, 2023

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From the journal Forum Mathematicum Volume 35 Issue 4

Abstract

In this paper, we study the joint distribution of the cokernels of random p-adic matrices. Let p be a prime and let P 1 ⁢ ( t ) , … , P l ⁢ ( t ) ∈ ℤ p ⁢ [ t ] be monic polynomials whose reductions modulo p in 𝔽 p ⁢ [ t ] are distinct and irreducible. We determine the limit of the joint distribution of the cokernels cok ⁡ ( P 1 ⁢ ( A ) ) , … , cok ⁡ ( P l ⁢ ( A ) ) for a random n × n matrix A over ℤ p with respect to the Haar measure as n → ∞ . By applying the linearization of a random matrix model, we also provide a conjecture which generalizes this result. Finally, we provide a sufficient condition that the cokernels cok ⁡ ( A ) and cok ⁡ ( A + B n ) become independent as n → ∞ , where B n is a fixed n × n matrix over ℤ p for each n and A is a random n × n matrix over ℤ p .

Keywords: Random p-adic matrix; Cohen–Lenstra heuristics

MSC 2020: 15B52; 60B20; 11C20

Communicated by Freydoon Shahidi

Funding statement: The author is supported by a KIAS Individual Grant (SP079601) via the Center for Mathematical Challenges at Korea Institute for Advanced Study.

Acknowledgements

We are very grateful to the anonymous referee for helpful comments and suggestions. We thank Gilyoung Cheong for his helpful comments.

References

[1] J. D. Achter, The distribution of class groups of function fields, J. Pure Appl. Algebra 204 (2006), no. 2, 316–333. 10.1016/j.jpaa.2005.04.003Search in Google Scholar

[2] M. Bhargava, D. M. Kane, H. W. Lenstra, Jr., B. Poonen and E. Rains, Modeling the distribution of ranks, Selmer groups, and Shafarevich–Tate groups of elliptic curves, Camb. J. Math. 3 (2015), no. 3, 275–321. 10.4310/CJM.2015.v3.n3.a1Search in Google Scholar

[3] I. Boreico, Statistics of random integral matrices, Ph.D. Thesis, Stanford University, 2016. Search in Google Scholar

[4] G. Cheong and Y. Huang, Cohen–Lenstra distributions via random matrices over complete discrete valuation rings with finite residue fields, Illinois J. Math. 65 (2021), no. 2, 385–415. 10.1215/00192082-8939615Search in Google Scholar

[5] G. Cheong and N. Kaplan, Generalizations of results of Friedman and Washington on cokernels of random p-adic matrices, J. Algebra 604 (2022), 636–663. 10.1016/j.jalgebra.2022.03.035Search in Google Scholar

[6] 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-xSearch in Google Scholar

[7] H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups of number fields, Number Theory (Noordwijkerhout 1983), Lecture Notes in Math. 1068, Springer, Berlin (1984), 33–62. 10.1007/BFb0099440Search in Google Scholar

[8] C. Delaunay, Heuristics on Tate–Shafarevitch groups of elliptic curves defined over ℚ , Experiment. Math. 10 (2001), no. 2, 191–196. 10.1080/10586458.2001.10504442Search in Google Scholar

[9] C. Delaunay, Heuristics on class groups and on Tate-Shafarevich groups: The magic of the Cohen–Lenstra heuristics, Ranks of Elliptic Curves and Random Matrix Theory, London Math. Soc. Lecture Note Ser. 341, Cambridge University, Cambridge (2007), 323–340. 10.1017/CBO9780511735158.021Search in Google Scholar

[10] C. Delaunay and F. Jouhet, p ℓ -torsion points in finite abelian groups and combinatorial identities, Adv. Math. 258 (2014), 13–45. 10.1016/j.aim.2014.02.033Search in Google Scholar

[11] E. Friedman and L. C. Washington, On the distribution of divisor class groups of curves over a finite field, Théorie des nombres (Quebec 1987), de Gruyter, Berlin (1989), 227–239. 10.1515/9783110852790.227Search in Google Scholar

[12] D. Garton, Random matrices, the Cohen–Lenstra heuristics, and roots of unity, Algebra Number Theory 9 (2015), no. 1, 149–171. 10.2140/ant.2015.9.149Search in Google Scholar

[13] F. Gerth, III, Densities for ranks of certain parts of p-class groups, Proc. Amer. Math. Soc. 99 (1987), no. 1, 1–8. 10.1090/S0002-9939-1987-0866419-3Search in Google Scholar

[14] M. Lipnowski, W. Sawin and J. Tsimerman, Cohen–Lenstra heuristics and bilinear pairings in the presence of roots of unity, preprint (2020), https://arxiv.org/abs/2007.12533. Search in Google Scholar

[15] H. H. Nguyen and M. M. Wood, Random integral matrices: Universality of surjectivity and the cokernel, Invent. Math. 228 (2022), no. 1, 1–76. 10.1007/s00222-021-01082-wSearch in Google Scholar

[16] A. Smith, The distribution of ℓ ∞ -Selmer groups in degree ℓ twist families, preprint (2022), https://arxiv.org/abs/2207.05674. Search in Google Scholar

[17] L. C. Washington, Some remarks on Cohen–Lenstra heuristics, Math. Comp. 47 (1986), no. 176, 741–747. 10.1090/S0025-5718-1986-0856717-9Search in Google Scholar

[18] M. M. Wood, The distribution of sandpile groups of random graphs, J. Amer. Math. Soc. 30 (2017), no. 4, 915–958. 10.1090/jams/866Search in Google Scholar

[19] M. M. Wood, Random integral matrices and the Cohen–Lenstra heuristics, Amer. J. Math. 141 (2019), no. 2, 383–398. 10.1353/ajm.2019.0008Search in Google Scholar

Received: 2022-07-19

Revised: 2023-03-13

Published Online: 2023-05-03

Published in Print: 2023-07-01

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https://doi.org/10.1515/forum-2022-0209

Keywords for this article

Random p-adic matrix; Cohen–Lenstra heuristics