On Some Exponential Sums of Conrey and Iwaniec

where the implied constant is absolute. This is proved in [1, §13, 14] (see also [5, Th. 11.42] for an outline of the proof). In this note, we sketch a different proof, based on the general philosophy of reduction to one-variable sums and on the use of the powerful form of Deligne’s proof of the Riemann Hypothesis over finite fields involving sums of trace functions of general sheaves [3] (see [4] for other recent systematic applications of this principle). Fix an auxiliary prime ` different from the characteristic of Fq, and a field-isomorphism ι : Q̄` ' C, which we use as an identification. For an `-adic sheaf F on some algebraic variety X/Fq, and some x ∈ X(Fq), we denote by tF,Fq(x) the value, under ι, of the trace function of F at the geometric Frobenius of Fq acting on the stalk at x. We have S(χ1, χ2) = ∑