Poincaré series and Kloosterman sums associated to the group SL(3, ℤ) were introduced and studied in (Bump, Friedberg and Goldfeld, 1988) following the point of view of Selberg (1965). A very nice exposition of the GL(2) theory is given in (Cogdell and Piatetski-Shapiro, 1990). The method was first generalized to GL(n) in (Friedberg, 1987), (Stevens, 1987). In (Bump, Friedberg and Goldfeld, 1988) it is shown that the SL(3, ℤ) Kloosterman sums are hyper Kloosterman sums associated to suitable algebraic varieties. Non-trivial bounds were obtained by using Hensel's lemma and Deligne's estimates for hyper-Kloosterman sums (Deligne, 1974) in (Larsen, 1988), and later (Dabrowski and Fisher, 1997) improved these bounds by also using methods from algebraic geometry following (Deligne, 1974). Sharp bounds for special types of Kloosterman sums were also obtained in (Friedberg, 1987a,c). In (Dabrowski, 1993), the theory of Kloosterman sums over Chevalley groups is developed. Important applications of the theory of GL(n) Kloosterman sums were obtained in (Jacquet, 2004b) (see also (Ye, 1998)).

Another fundamental direction for research in the theory of Poincaré series and Kloosterman sums was motivated by the GL(2) Kuznetsov trace formula, (see (Kuznecov, 1980) and also (Bruggeman, 1978)). Generalizations of the Kuznetsov trace formula to GL(n), with n ≥ 3 were obtained in (Friedberg, 1987), (Goldfeld, 1987), (Ye, 2000), but they have not yet proved useful for analytic number theory. The chapter concludes with a new version of the GL(n) Kuznetsov trace formula derived by Xiaoqing Li.