Reference to "bounds of Weil and Deligne" In the this paper by Friedlander and Iwaniec, it is said that they are "able to avoid much of the high-powered technology frequently used in modern analytic number theory such as the bounds of Weil and Deligne." What bounds are being referred to here?
 A: The first of these bounds was explicitly stated by Weil in the short paper


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*André Weil, "On some exponential sums" (1948)


It depends on the Riemann hypothesis for curves over finite fields, which he had proved back in the early 40s, and shows that
$$|S(m,n;p)|\leq 2 \sqrt{p}$$
where $S(m,n;p)$ is the classic Kloosterman sum
$$S(m,n;p)=\sum_{\substack{{x (\mathrm{mod}\, p)} \\ (x,p)=1}} e\left(\frac{mx+nx^{-1}}{p}\right)$$
The second bound is due to Deligne, and holds for the more general hyper-Kloosterman sum, that is, sums of the form
$$K_n(p)=\sum_{\substack{{x_1,...,x_n (\mathrm{mod}\, p)} \\ x_1 \cdots x_n=1}} e\left(\frac{x_1+\cdots+x_n}{p}\right)$$
and the precise estimate is
$$|K_n(p)|\leq np^{(n-1)/2}$$
This one depends on the Riemann hypothesis for functions fields, which Deligne had proved on the famous paper


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*Pierre Deligne, "La conjecture de Weil: I" (1974)


The applications to hyper-Kloosterman sum were worked out in detail in


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*Pierre Deligne, "SGA 4½: Cohomologie étale" (1977)


particularly the Exposé 6, "Applications de la formule des traces aux sommes trigonométriques", section 7, "Sommes de Kloosterman généralisées".
It also important to mention the work of Katz on exponential sums, building directly on that of Deligne, particuarly the book


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*Nicholas Katz "Gauss Sums, Kloosterman Sums, and Monodromy Groups" (1988)

