Pairs of commuting integer matrices Browning, Timothy D ; https://orcid.org/0000-0002-8314-0177 Sawin, Will Wang, Victor ; https://orcid.org/0000-0002-0704-7026 ddc:510 We prove upper and lower bounds on the number of pairs of commuting n x n matrices with integer entries in [-T, T], as T -> . Our work uses Fourier analysis and leads to an analysis of exponential sums involving matrices over finite fields. These are bounded by combining a stratification result of Fouvry and Katz with a new result about the flatness of the commutator Lie bracket. Springer Nature 2025 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/20367 https://research-explorer.ista.ac.at/download/20367/20950 Browning TD, Sawin W, Wang V. Pairs of commuting integer matrices. <i>Mathematische Annalen</i>. 2025;393:1863–1880. doi:<a href="https://doi.org/10.1007/s00208-025-03285-5">10.1007/s00208-025-03285-5</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s00208-025-03285-5 info:eu-repo/semantics/altIdentifier/issn/0025-5831 info:eu-repo/semantics/altIdentifier/e-issn/1432-1807 info:eu-repo/semantics/altIdentifier/wos/001567740200001 info:eu-repo/semantics/altIdentifier/arxiv/2409.01920 https://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess