Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Selberg trace formula for $ {\rm SL}(3,{\bf Z})\backslash{\rm SL}(3,{\bf R})/{\rm SO}(3,{\bf R})$


Author: D. I. Wallace
Journal: Trans. Amer. Math. Soc. 345 (1994), 1-36
MSC: Primary 11F72
DOI: https://doi.org/10.1090/S0002-9947-1994-1184117-4
MathSciNet review: 1184117
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we compute the trace formula for $ SL(3,\mathbb{Z})$ in detail and refine it to a greater extent than has previously been done. We show that massive cancellation occurs in the parabolic terms, leading to a far simpler formula than had been thought possible.


References [Enhancements On Off] (What's this?)

  • [1] J. Arthur, The trace formula for compact quotient, Proc. Internat. Congress Math., Warsaw, 1983. MR 723181 (85j:22031)
  • [2] -, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), 1-74. MR 625344 (84a:10031)
  • [3] -, A trace formula for reductive groups. I, Duke Math. J. 45 (1978), 911-953. MR 518111 (80d:10043)
  • [4] -, A trace formula for reductive groups. II: Applications of a truncation operator, Compositio Math. 40 (1980), 87-121. MR 558260 (81b:22018)
  • [5] R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, Wiley-Interscience, New York, 1961.
  • [6] I. Efrat, The Selberg trace formula for $ PS{L_2}{(\mathbb{R})^n}$. Mem. Amer. Math. Soc., vol. 65, 1987. MR 874084 (88e:11041)
  • [7] Y. Z. Flicker, The trace formula and base change for $ GL(3)$, Lecture Notes in Math., vol. 927, Springer-Verlag, Berlin and New York, 1982. MR 663002 (84d:10035)
  • [8] I. M. Gelfand, M. I. Graev, and I. I. Piatetski-Shapiro, Representation theory and automorphic functions, Saunders, Philadelphia, PA, 1966.
  • [9] D. Hejhal, The Selberg trace formula for $ PSL(2,\mathbb{R})$, vol. I, Lecture Notes in Math., vol. 548, Springer-Verlag, Berlin and New York, 1976. MR 0439755 (55:12641)
  • [10] -, The Selberg trace formula for $ PSL(2,\mathbb{R})$, vol. II, Lecture Notes in Math., vol. 1001, Springer-Verlag, Berlin and New York, 1983.
  • [11] S. Helgason, Groups and geometric analysis, Academic Press, San Diego, 1984. MR 754767 (86c:22017)
  • [12] K. Imai and A. Terras, Fourier expansions of Eisenstein series for $ GL(3,\mathbb{Z})$, Trans. Amer. Math. Soc. 273 (1982), 679-694. MR 667167 (84d:10033)
  • [13] J. A. C. Kolk, The Selberg trace formula and asymptotic behavior of spectra, Ph.D. thesis, Rigksuniversiteit te Utrecht, 1977. MR 476489 (80a:22017)
  • [14] T. Kubota, Elementary theory of Eisenstein series, Halsted Press, New York, 1973. MR 0429749 (55:2759)
  • [15] S. Lang, $ S{L_2}(\mathbb{R})$, Addison-Wesley, Reading, Mass., 1975. MR 0430163 (55:3170)
  • [16] R. Langlands, Eisenstein series, Proc. Sympos. Pure Math., vol. 9, Amer. Math. Soc., Providence, RI, 1966. MR 0249539 (40:2784)
  • [17] W. Müller, The trace class conjecture in the theory of automorphic forms, Ann. of Math. (2) 130 (1989), 473-529. MR 1025165 (90m:11083)
  • [18] S. Osborne and G. Warner, The Selberg trace formula. I: $ \Gamma $-rank one lattices, Crelles J. 324 (1981), 1-113. MR 614517 (83m:10044)
  • [19] -, The Selberg trace formula. II: Partition reduction, truncation, Pacific J. Math. 106 (1983), 307-496. MR 699915 (87c:22022a)
  • [20] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 115 (1982), 229-247. MR 675187 (84i:10023a)
  • [21] H. Schlichtkrull, Hyperfunctions and harmonic analysis on symmetric spaces, Birkhäuser, Boston and Basel, 1984. MR 757178 (86g:22021)
  • [22] A. Selberg, Lectures on the trace formula, Univ. of Gottingen, 1954.
  • [23] A. Terras, Harmonic analysis on symmetric spaces and applications. I, Springer-Verlag, Berlin and New York, 1985. MR 791406 (87f:22010)
  • [24] -, Harmonic analysis on symmetric spaces and applications. II, Springer-Verlag, Berlin and New York, 1988. MR 955271 (89k:22017)
  • [25] A. B. Venkov, The Selberg trace formula for $ SL(3,\mathbb{Z})$, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI) 37 (1973).
  • [26] D. I. Wallace, Conjugacy classes of hyperbolic matrices and ideal classes in an order, Trans. Amer. Math. Soc. 283 (1984), 177-184. MR 735415 (85h:11024)
  • [27] -, Explicit form of the hyperbolic term in the Selberg trace formula for $ SL(3,\mathbb{Z})$, J. Number Theory 24 (1986), 127-133. MR 863649 (88c:11029)
  • [28] -, A preliminary version of the Selberg trace formula for $ SL(3,\mathbb{Z})\backslash SL(3,\mathbb{R})/SO(3,\mathbb{R})$, Contemp. Math. 53 (1986), 11-15.
  • [29] -, Maximal parabolic terms in the Selberg trace formula for $ SL(3,\mathbb{Z})\backslash SL(3,\mathbb{R})/SO(3,\mathbb{R})$, J. Number Theory 32 (1959). MR 945590 (89m:11053)
  • [30] -, Terms in the Selberg trace formula for $ SL(3,\mathbb{Z})\backslash SL(3,\mathbb{R})/SO(3,\mathbb{R})$, associated to Eisenstein series coming from a maximal parabolic subgroup, Proc. Amer. Math. Soc. 106 (1989), 875-883. MR 963577 (90e:11081)
  • [31] -, Terms in the Selberg trace formula for $ SL(3,\mathbb{Z})\backslash SL(3,\mathbb{R})/SO(3,\mathbb{R})$, associated to Eisenstein series coming from a minimal parabolic subgroup, Trans. Amer. Math. Soc. (to appear). MR 1031979 (92a:11062)
  • [32] -, The loxodromic term of the Selberg trace formula for $ SL(3,\mathbb{Z})\backslash SL(3,\mathbb{R})/SO(3,\mathbb{R})$ (to appear).
  • [33] -, Minimal parabolic terms in the Selberg trace formula for $ SL(3,\mathbb{Z})\backslash SL(3,\mathbb{R})/SO(x,\mathbb{R})$, J. Number Theory 32 (1989), 1-13. MR 1002111 (91b:11066)
  • [34] G. Warner, Selberg's trace formula for non-uniform lattices. The R-rank one case, Adv. Math. Stud. 6, 1-142. MR 535763 (81f:10044)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11F72

Retrieve articles in all journals with MSC: 11F72


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1184117-4
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society