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is that they can be used to express the Fourier coefficients of the Poincaré series for the group GL(2,Q). As Kloosterman [Kl] pointed out, estimates of the Kloosterman sums lead to bounds for the Fourier coefficients of modular forms (see also Selberg [Se]). To estimate S(a, b; c) one easily reduces to the case c = p and p ab, with p prime. Salié [Sa] explicitly calculated S(a, b; p) when m > 1 and Weil [W1] proved that |S(a, b; p)| ≤ 2√p as a consequence of his proof of the Riemann hypothesis for curves. Thanks to Deligne [D], we now have efficient techniques for estimating exponential sums such as S(a, b; p). Paradoxically, the simpler case of exponential sums over Z/pZ with m > 1 is in some ways less well understood. Smith and Loxton [Sm1, Sm2, Lo-Sm1] generalized Salié’s methods and Katz [K1] interpreted such results as a stationary phase formula. We take Katz’s point of view, proving and generalizing his statement in Section 1. Our statement is very convenient for applications, and in many cases it gives sharper bounds than those of Smith and Loxton. We give several examples to illustrate the use of our theorem. The theory of Poincaré series for GL(3,Q) was developed by Bump, Friedberg, and Goldfeld [B-F-G] and extended to GL(N,Q) independently by Friedberg [F] and Stevens [S]. The Fourier coefficients of these Poincaré

A stationary phase formula for exponential sums over $ℤ/p^{m}ℤ$ and applications to GL(3)-Kloosterman sums