The Selberg trace formula for 𝑆𝐿(3,𝑍)\𝑆𝐿(3,𝑅)/𝑆𝑂(3,𝑅)

Wallace, D. I.

doi:10.1090/S0002-9947-1994-1184117-4

The Selberg trace formula for $\textrm {SL}(3,\textbf {Z})\backslash \textrm {SL}(3,\textbf {R})/\textrm {SO}(3,\textbf {R})$
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Trans. Amer. Math. Soc. 345 (1994), 1-36 Request permission

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

James Arthur, The trace formula for reductive groups, Conference on automorphic theory (Dijon, 1981) Publ. Math. Univ. Paris VII, vol. 15, Univ. Paris VII, Paris, 1983, pp. 1–41. MR 723181, DOI 10.1007/978-1-4684-6730-7_{1}
James Arthur, The trace formula in invariant form, Ann. of Math. (2) 114 (1981), no. 1, 1–74. MR 625344, DOI 10.2307/1971376
James G. Arthur, A trace formula for reductive groups. I. Terms associated to classes in $G(\textbf {Q})$, Duke Math. J. 45 (1978), no. 4, 911–952. MR 518111
James Arthur, A trace formula for reductive groups. II. Applications of a truncation operator, Compositio Math. 40 (1980), no. 1, 87–121. MR 558260

Methods of mathematical physics

Isaac Y. Efrat, The Selberg trace formula for $\textrm {PSL}_2(\textbf {R})^n$, Mem. Amer. Math. Soc. 65 (1987), no. 359, iv+111. MR 874084, DOI 10.1090/memo/0359
Yuval Z. Flicker, The trace formula and base change for $\textrm {GL}(3)$, Lecture Notes in Mathematics, vol. 927, Springer-Verlag, Berlin-New York, 1982. MR 663002, DOI 10.1007/BFb0094272

Representation theory and automorphic functions

Dennis A. Hejhal, The Selberg trace formula for $\textrm {PSL}(2,R)$. Vol. I, Lecture Notes in Mathematics, Vol. 548, Springer-Verlag, Berlin-New York, 1976. MR 0439755, DOI 10.1007/BFb0079608

The Selberg trace formula for

Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
K. Imai and A. Terras, The Fourier expansion of Eisenstein series for $\textrm {GL}(3,\,\textbf {Z})$, Trans. Amer. Math. Soc. 273 (1982), no. 2, 679–694. MR 667167, DOI 10.1090/S0002-9947-1982-0667167-5
Johan Antoon Casper Kolk, The Selberg trace formula and asymptotic behaviour of spectra, Rijksuniversiteit te Utrecht, Utrecht, 1977. With a Dutch summary; Doctoral dissertation, University of Utrecht. MR 0476489
Tomio Kubota, Elementary theory of Eisenstein series, Kodansha, Ltd., Tokyo; Halsted Press [John Wiley & Sons, Inc.], New York-London-Sydney, 1973. MR 0429749
Serge Lang, $\textrm {SL}_{2}(\textbf {R})$, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. MR 0430163
R. P. Langlands, Eisenstein series, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 235–252. MR 0249539
Werner Müller, The trace class conjecture in the theory of automorphic forms, Ann. of Math. (2) 130 (1989), no. 3, 473–529. MR 1025165, DOI 10.2307/1971453
M. Scott Osborne and Garth Warner, The Selberg trace formula. I. $\Gamma$-rank one lattices, J. Reine Angew. Math. 324 (1981), 1–113. MR 614517, DOI 10.1515/crll.1981.324.1
M. Scott Osborne and Garth Warner, The Selberg trace formula. II. Partition, reduction, truncation, Pacific J. Math. 106 (1983), no. 2, 307–496. MR 699915, DOI 10.2140/pjm.1983.106.307
Peter Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982), no. 2, 229–247. MR 675187, DOI 10.1016/0022-314X(82)90028-2
Henrik Schlichtkrull, Hyperfunctions and harmonic analysis on symmetric spaces, Progress in Mathematics, vol. 49, Birkhäuser Boston, Inc., Boston, MA, 1984. MR 757178, DOI 10.1007/978-1-4612-5298-6

Lectures on the trace formula

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, Harmonic analysis on symmetric spaces and applications. II, Springer-Verlag, Berlin, 1988. MR 955271, DOI 10.1007/978-1-4612-3820-1

The Selberg trace formula for

37

D. I. Wallace, Conjugacy classes of hyperbolic matrices in $\textrm {Sl}(n,\,\textbf {Z})$ and ideal classes in an order, Trans. Amer. Math. Soc. 283 (1984), no. 1, 177–184. MR 735415, DOI 10.1090/S0002-9947-1984-0735415-0
D. I. Wallace, Explicit form of the hyperbolic term in the Selberg trace formula for $\textrm {Sl}(3,\textbf {Z})$ and Pell’s equation for hyperbolics in $\textrm {Sl}(3,\textbf {Z})$, J. Number Theory 24 (1986), no. 2, 127–133. MR 863649, DOI 10.1016/0022-314X(86)90097-1

A preliminary version of the Selberg trace formula for

53

D. I. Wallace, Maximal parabolic terms in the Selberg trace formula for $\textrm {SL}(3,\textbf {Z})\backslash \textrm {SL}(3,\textbf {R})/\textrm {SO}(3,\textbf {R})$, J. Number Theory 29 (1988), no. 2, 101–117. MR 945590, DOI 10.1016/0022-314X(88)90095-9
D. I. Wallace, Terms in the Selberg trace formula for $\textrm {SL}(3,\textbf {Z})\backslash \textrm {SL}(3,\textbf {R})/\textrm {SO}(3,\textbf {R})$ associated to Eisenstein series coming from a maximal parabolic subgroup, Proc. Amer. Math. Soc. 106 (1989), no. 4, 875–883. MR 963577, DOI 10.1090/S0002-9939-1989-0963577-9
D. I. Wallace, Terms in the Selberg trace formula for $\textrm {SL}(3,{\scr Z})\backslash \textrm {SL}(3,{\scr R})/\textrm {SO}(3,{\scr R})$ associated to Eisenstein series coming from a minimal parabolic subgroup, Trans. Amer. Math. Soc. 327 (1991), no. 2, 781–793. MR 1031979, DOI 10.1090/S0002-9947-1991-1031979-4

The loxodromic term of the Selberg trace formula for

D. I. Wallace, Minimal parabolic terms in the Selberg trace formula for $\textrm {SL}(3,\textbf {Z})\backslash \textrm {SL}(3,\textbf {R})/\textrm {SO}(3,\textbf {R})$, J. Number Theory 32 (1989), no. 1, 1–13. MR 1002111, DOI 10.1016/0022-314X(89)90094-2
Garth Warner, Selberg’s trace formula for nonuniform lattices: the $R$-rank one case, Studies in algebra and number theory, Adv. in Math. Suppl. Stud., vol. 6, Academic Press, New York-London, 1979, pp. 1–142. MR 535763