{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:09:54Z","timestamp":1760144994105,"version":"build-2065373602"},"reference-count":10,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T00:00:00Z","timestamp":1717459200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fields Institute through the Fields Undergraduate Summer Research Program in 2021"},{"name":"Natural Sciences and Engineering Research Council of Canada"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we prove a discrete analog of the Selberg Trace Formula for the group GL3(Fq). By considering a cubic extension of the finite field Fq, we define an analog of the upper half-space and an action of GL3(Fq) on it. To compute the orbital sums, we explicitly identify the double coset spaces and fundamental domains in our upper half space. To understand the spectral side of the trace formula, we decompose the induced representation \u03c1=Ind\u0393G1 for G=GL3(Fq) and \u0393=GL3(Fp).<\/jats:p>","DOI":"10.3390\/axioms13060381","type":"journal-article","created":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T07:39:49Z","timestamp":1717486789000},"page":"381","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Selberg Trace Formula for GL3(Fp)\u2216GL3(Fq)\/K"],"prefix":"10.3390","volume":"13","author":[{"given":"Daksh","family":"Aggarwal","sequence":"first","affiliation":[{"name":"Department of Mathematics, Brown University, Providence, RI 02912, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8281-8321","authenticated-orcid":false,"given":"Asghar","family":"Ghorbanpour","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2552-0208","authenticated-orcid":false,"given":"Masoud","family":"Khalkhali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiyuan","family":"Lu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-2163-0365","authenticated-orcid":false,"given":"Bal\u00e1zs","family":"N\u00e9meth","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C Shijia","family":"Yu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,4]]},"reference":[{"key":"ref_1","first-page":"47","article-title":"Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series","volume":"20","author":"Selberg","year":"1956","journal-title":"J. Indian Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/0022-314X(82)90028-2","article-title":"Class numbers of indefinite binary quadratic forms","volume":"15","author":"Sarnak","year":"1982","journal-title":"J. Number Theory"},{"key":"ref_3","first-page":"1","article-title":"The Selberg trace formula for SL(3,Z)\u2216SL(3,R)\/SO(3,R)","volume":"345","author":"Wallace","year":"1994","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_4","unstructured":"Arthur, J. (2003, January 2\u201327). An introduction to the trace formula. Proceedings of the Clay Mathematics Proceedings, Toronto, ON, Canada."},{"key":"ref_5","first-page":"709","article-title":"Fonctions p\u00e9riodiques et formule des traces de selberg sur les arbres","volume":"305","author":"Ahumada","year":"1987","journal-title":"CR Acad. Sci. Paris"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Terras, A. (1999). Fourier Analysis on Finite Groups and Applications, Cambridge University Press. London Mathematical Society Student Texts.","DOI":"10.1017\/CBO9780511626265"},{"key":"ref_7","first-page":"33","article-title":"Ramanujan graphs","volume":"18","author":"Murty","year":"2003","journal-title":"J. Ramanujan Math. Soc."},{"key":"ref_8","first-page":"105","article-title":"On the conjugacy classes of the general linear group GL(3,q)","volume":"38","author":"Hering","year":"1994","journal-title":"Bull. Math. Soc. Sci. Math. Roum."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1006\/jnth.2000.2524","article-title":"The finite upper half space and related hypergraphs","volume":"84","author":"Martinez","year":"2000","journal-title":"J. Number Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"225","DOI":"10.4153\/CJM-1951-027-x","article-title":"The representations of gl(3,q), gl(4,q), pgl(3,q), and pgl(4,q)","volume":"3","author":"Steinberg","year":"1951","journal-title":"Can. J. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/381\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:53:35Z","timestamp":1760108015000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/381"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,4]]},"references-count":10,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["axioms13060381"],"URL":"https:\/\/doi.org\/10.3390\/axioms13060381","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,6,4]]}}}