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A Survey on Pure and Mixed Exponential Sums Modulo Prime Powers

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A Survey on Pure and Mixed Exponential Sums Modulo Prime Powers A SURVEY ON PURE AND MIXED EXPONENTIAL SUMS MODULO PRIME POWERS TODD COCHRANE AND ZHIYONG ZHENG 1. Introduction In this paper we give a survey of recent work by the authors and others on pure and mixed exponential sums of the type pm pm X X epm (f(x)); χ(g(x))epm (f(x)); x=1 x=1 m 2πix=pm where p is a prime power, epm (·) is the additive character epm (x) = e and χ is a multiplicative character (mod pm). The goals of this paper are threefold; first, to point out the similarity between exponential sums over finite fields and exponential sums over residue class rings (mod pm) with m ≥ 2; second, to show how mixed exponential sums can be reduced to pure exponential sums when m ≥ 2 and third, to make a thorough review of the formulae and upper bounds that are available for such sums. Included are some new observations and consequences of the methods we have developed as well as a number of open questions, some very deep and some readily accessible, inviting the reader to a further investigation of these sums. 2. Pure Exponential Sums We start with a discussion of pure exponential sums of the type q X (2.1) S(f; q) = eq(f(x)) x=1 2πif(x)=q where f(x) is a polynomial over Z, q is any positive integer and eq(f(x)) = e . These sums enjoy the multiplicative property k Y mi S(f; q) = S(λif; pi ) i=1 Qk mi Pk mi where q = i=1 pi and i=1 λiq=pi = 1, reducing their evaluation to the case of prime power moduli. For the case of prime moduli we have the fundamental result of Weil [92] that for any polynomial f(x) of degree d not divisible by p, there exists a set of d − 1 complex numbers p !1;:::!d−1, each of modulus p such that (2.2) S(f; p) = !1 + !2 + ··· + !d−1: Date: September 8, 2009. 1991 Mathematics Subject Classification. 11L07;11L03. Key words and phrases. exponential sums. The first author wishes to thank Tsinghua University, Beijing, P.R.C., and the National Center for Theoretical Sciences, Hsinchu, Taiwan, for hosting his visits and supporting this work. The second author was supported by the N.S.F. of the P.R.C. for distinguished young scholars. 1 2 TODD COCHRANE AND ZHIYONG ZHENG Moreover, if the sum is extended to the finite field Fpj by setting X Sj = ep(T rj (f(x)); x2 Fpj j j where T rj is the trace mapping from Fpj into Fp, then Sj = !1 +···+!d−1. These results follow from the rationality of the L-function associated with the sums Sj , together with the accompanying Riemann Hypothesis. In Weil's original work [92] it is only shown that the number of characteristic values !i is less than or equal to d − 1. Bombieri [3] proved the number actually equals d − 1. The reader is referred to the works of Schmidt [82] and Stepanov [90] for elementary proofs of these results. It is perhaps less well known that there is a striking analogy for prime power moduli pm with m ≥ 2. To illustrate this let t be the largest power of p dividing all of the coefficients of f 0(x). Note t = 0 if p > deg(f). We define a (2.3) Critical Point Congruence: C(x) := p−tf 0(x) ≡ 0 (mod p); together with a set of (2.4) Critical Points: A = fα 2 Fp : C(α) ≡ 0 (mod p)g: To evaluate S(f; pm) it is convenient to break up the sum over the different residue m Pp m classes (mod p), S(f; p ) = α=1 Sα(f; p ) where pm m X (2.5) Sα(f; p ) = epm (f(x)): x=1 x≡α (mod p) Upon setting x = u + pm−1v with u running from 1 to pm−1, subject to the constraint u ≡ α (mod p), and v running from 1 to p, one readily sees that if p is odd and m ≥ t + 2 or p = 2 and m ≥ 3 then Sα = 0 unless α is a critical point, and so m X m (2.6) S(f; p ) = Sα(f; p ): α2A In particular, if there are no critical points then the sum is zero. If α is a critical point of multiplicity one then for odd p and m ≥ t + 2, 8 ∗ m+t m 2 m <ep (f(α ))p ; if m − t is even; (2.7) Sα(f; p ) = A ∗ m+t−1 α m 2 : p ep (f(α ))p Gp; if m − t is odd; where α∗ is the unique lifting of α to a solution of the congruence p−tf 0(x) ≡ 0 (mod p[(m−t+1)=2]); −t 00 · Aα ≡ 2p f (α) (mod p), ( p ) is the Legendre symbol and Gp is the quadratic Gauss sum p−1 (p X 2 p; if p ≡ 1 (mod 4); Gp := ep(x ) = p x=0 i p; if p ≡ 3 (mod 4): A similar formula holds for p = 2; see [12] section 5. It follows from (2.6) and (2.7) that if p is odd, m ≥ t + 2 and all critical points are of multiplicity one then m 8 p ∗ m+t P m 2 X < α2A ep (f(α ))p ; if m − t is even; m (2.8) ep (f(x)) = A ∗ m+t−1 P α m 2 x=1 : α2A p ep (f(α ))p Gp if m − t is odd. It is not known to the authors when this formula may have first appeared in the literature but the technique dates back at least to the work of Sali´e[80]. Katz [47, p. 110] and Dabrowski and Fisher [17] have stated generalizations of this formula in higher dimensions and refer to such formulae as stationary phase formulae. In the result of Dabrowski and EXPONENTIAL SUMS 3 Fisher it is also allowed that the critical points have multiplicity greater than one, but the result obtained in this case is less explicit. The authors derived the formula as stated above in their work [12]. To illustrate the formula consider the quadratic Gauss sum S(Ax2; pm), with p odd, p - A. The critical point congruence is just 2Ax ≡ 0 (mod p), and so t = 0 and there is a single critical point α = 0 of multiplicity one. Thus ( m=2 2 m p ; if m is even; S(Ax ; p ) = m−1 A 2 p Gpp ; if m is odd: This of course is a well known formula due to Gauss. The formula in (2.7) holds as well for rational functions; see Katz [47], Dabrowski and Fisher [17] or the author's work [13]. For the Kloosterman sum S(Ax + Bx−1; pm), with 2 AB p - AB, the critical point congruence is Ax − B ≡ 0 (mod p). If p = −1 then the AB sum is zero for m ≥ 2, while if p = 1 then there are two critical points of multiplicity one and (2.8) yields the classical formula of Sali´e[80], proven also by Whiteman [93], Estermann [23], Carlitz [4] and Williams [94]. The extra factor t=2 occuring in the exponent on the right-hand side of (2.8) may be understood by realizing that S(f; pm) actually degenerates to a complete sum (mod pm−t) in the following sense, pm−t m t X (2.9) S(f; p ) = p epm (f(x)): x=1 The latter sum may be regarded as a nonstandard exponential sum (mod pm−t) since f(x) ≡ f(y) (mod pm) if x ≡ y (mod pm−t). By (2.8), it may be expressed as a sum of m−t complex numbers of moduli p 2 . It is quite straightforward to obtain the formula in (2.7). For instance, if m is even and t = 0 then write x = u + pm=2v with u and v running from 1 to pm=2, and note that f(x) ≡ f(u) + f 0(u)pm=2v (mod pm). Thus m X X 0 m=2 X S(f; p ) = epm (f(u)) epm=2 (f (u)v) = p epm (f(u)); u v pm=2jf 0(u) which is the stated result. Since there are at most d − 1 critical points, we see from (2.8) that S(f; pm) can be expressed as a sum of at most d − 1 complex numbers of moduli m+t p 2 , and moreover the values of these numbers are explicit. Question 1. Is there a general formula for the values !i in (2.2) for (mod p) exponential sums? This is a very deep and unyielding problem. For the case of Gauss sums S(Axd; p) formulae (sometimes with sign ambiguities) exist for a number of small values of d. The reader is referred to the book of Berndt, Evans and Williams [2] for a report on progress that has been made for Gauss sums. The value of the Kloosterman sum also is unknown when m = 1, although it is known that it may be expressed as a sum of two complex numbers of p moduli p. The formula obtained above for m ≥ 2 unfortunately cannot be extrapolated to the case m = 1. Curiously, if the Kloosterman sum is modified by inserting the Legendre Pp x B symbol to create a Sali´esum, x=1 p ep(Ax + x ), then it can be evaluated, and the formula one obtains is the extrapolation of the formula for odd m ≥ 3 that we state in (7.7).
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