Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Hecke operators and the fundamental domain for $\textrm {SL}(3,\textbf {Z})$
HTML articles powered by AMS MathViewer

by Daniel Gordon, Douglas Grenier and Audrey Terras PDF
Math. Comp. 48 (1987), 159-178 Request permission

Abstract:

We report on a detailed study of the fundamental domain for the special linear group ${\text {SL}}(3,{\mathbf {Z}})$ of $3 \times 3$ integral matrices with determinant one. Graphs of points coming from the action of Hecke operators are considered.
References
  • Emil Artin, The collected papers of Emil Artin, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965. Edited by Serge Lang and John T. Tate. MR 0176888, DOI 10.1007/978-1-4614-6294-1
  • Daniel Bump, Automorphic forms on $\textrm {GL}(3,\textbf {R})$, Lecture Notes in Mathematics, vol. 1083, Springer-Verlag, Berlin, 1984. MR 765698, DOI 10.1007/BFb0100147
  • Erhard Gottschling, Explizite Bestimmung der RandflĂ€chen des Fundamentalbereiches der Modulgruppe zweiten Grades, Math. Ann. 138 (1959), 103–124 (German). MR 107020, DOI 10.1007/BF01342938
  • D. Grenier, Fundamental Domains for ${P_n}{\text {/GL}}(n,{\mathbf {Z}})$, Ph.D. Thesis, Univ. of California at San Diego, June, 1986.
  • Martin C. Gutzwiller, Physics and Selberg’s trace formula, The Selberg trace formula and related topics (Brunswick, Maine, 1984) Contemp. Math., vol. 53, Amer. Math. Soc., Providence, RI, 1986, pp. 215–251. MR 853561, DOI 10.1090/conm/053/853561
  • A. Korkine and G. Zolotareff, Sur les formes quadratiques, Math. Ann. 6 (1873), no. 3, 366–389 (French). MR 1509828, DOI 10.1007/BF01442795
  • A. Lubotzky, R. Phillips, and P. Sarnak, Hecke operators and distributing points on the sphere. I, Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S149–S186. Frontiers of the mathematical sciences: 1985 (New York, 1985). MR 861487, DOI 10.1002/cpa.3160390710
  • Hans Maass, Siegel’s modular forms and Dirichlet series, Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin-New York, 1971. Dedicated to the last great representative of a passing epoch. Carl Ludwig Siegel on the occasion of his seventy-fifth birthday. MR 0344198, DOI 10.1007/BFb0058625
  • Hans Maass, Die Primzahlen in der Theorie der Siegelschen Modulfunktionen, Math. Ann. 124 (1951), 87–122 (German). MR 47075, DOI 10.1007/BF01343553
  • H. Minkowski, Gesammelte Abhandlungen, Chelsea, New York, 1911 (reprinted, 1967). N. V. Novikova, "Korkin-Zolotarev reduction domains of positive quadratic forms in $n \leqslant 8$ variables and a reduction algorithm for these domains," Soviet Math. Dokl., v. 27, 1983, pp. 557-560. S. S. RyĆĄkov, "The theory of Hermite-Minkowski reduction of positive quadratic forms," J. Soviet Math., v. 6, 1976, pp. 651-676. S. S. RyĆĄkov & E. P. Baranovskiǐ, "Classical methods in the theory of lattice packings," Russian Math. Surveys, v. 34, no. 4, 1979, pp. 1-68.
  • Peter Sarnak, Statistical properties of eigenvalues of the Hecke operators, Analytic number theory and Diophantine problems (Stillwater, OK, 1984) Progr. Math., vol. 70, BirkhĂ€user Boston, Boston, MA, 1987, pp. 321–331. MR 1018385
  • Goro Shimura, Introduction to the arithmetic theory of automorphic functions, KanĂŽ Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Publications of the Mathematical Society of Japan, No. 11. MR 0314766
  • H. M. Stark, Fourier coefficients of Maass waveforms, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 263–269. MR 803370
  • Audrey Terras, Harmonic analysis on symmetric spaces and applications. I, Springer-Verlag, New York, 1985. MR 791406, DOI 10.1007/978-1-4612-5128-6
  • Audrey Terras, On automorphic forms for the general linear group, Rocky Mountain J. Math. 12 (1982), no. 1, 123–143. MR 649746, DOI 10.1216/RMJ-1982-12-1-123
  • B. Wenkov, Über die Reduction positiver quadratischer Formen, Bull. Acad. Sci. URSS. SĂ©r. Math. [Izvestia Akad. Nauk SSSR] 4 (1940), 37–52 (Russian, with German summary). MR 0003001
  • Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, BirkhĂ€user Verlag, Basel, 1984. MR 776417, DOI 10.1007/978-1-4684-9488-4
Similar Articles
Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Math. Comp. 48 (1987), 159-178
  • MSC: Primary 11F25; Secondary 11F55, 11Y35
  • DOI: https://doi.org/10.1090/S0025-5718-1987-0866106-X
  • MathSciNet review: 866106