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 √ ω1, . . . ωd−1, each of modulus p such that

(2.2) S(f, p) = ω1 + ω2 + ··· + ωd−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)), x∈ Fpj j j where T rj is the trace mapping from Fpj into Fp, then Sj = ω1 +···+ωd−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 ωi 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 = {α ∈ Fp : C(α) ≡ 0 (mod p)}. 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 ). α∈A 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,  ∗ 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 (√ X 2 p, if p ≡ 1 (mod 4); Gp := ep(x ) = √ 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  p ∗ m+t P m 2 X  α∈A 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  α∈A 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/2|f 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 ωi 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 √ 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 Saliésum, 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). In general, however, it appears that the set of critical points does not play any role in the evaluation of (mod p) exponential sums. Question 2. In what cases does the formula stated here for m ≥ 2 hold as well for the prime field case m = 1? Is this coincidence a sporadic phenomenon? 4 TODD COCHRANE AND ZHIYONG ZHENG

Question 3. Is there a uniﬁed theory of exponential sums that yields both the results of Weil for the case m = 1 and the formula stated here for m ≥ 2? This question becomes perhaps more compelling when one realizes that the same analogy holds between mixed exponential sums with prime moduli and with prime power moduli; see (7.7). There are two shortcomings to the formula in (2.8). The ﬁrst is the case m = t + 1; m m−1 (we may trivially assume m > t by reducing the degree of f via the identity xp ≡ xp (mod pm) ). In this case S(f, pm) degenerates to a nonstandard (mod p) exponential Pp2 p Pp p sum. A well known example here is the Heilbronn sum x=1 ep2 (x ) = p x=1 ep2 (x ). Nontrivial estimates for this sum have been obtained by Heath-Brown [28] and Heath- Pp p 7/8 Brown and Konyagin [29], the latter being | x=1 ep2 (x )| p . Question 4. Does there exist in general a nontrivial bound for S(f, pm) when m = t + 1? The second limitation of the formula occurs when there is a critical point of multiplicity greater than one. Consider for example the Gauss sum S(Axd, pm) with p > d ≥ 3 and m ≥ 2. For this sum there is a single critical point x = 0 of multiplicity d−1 > 1, and thus d m d m the formula above doesn’t apply. However, by (2.6) we see that S(Ax , p ) = S0(Ax , p ), that is the only contribution to the sum comes from values of x ≡ 0 (mod p). We obtain S(Axd, pm) = S(Apdxd, pm−1) = pd−1S(Axd, pm−d). The argument can be repeated provided m − d ≥ 2. In particular, if d|m then we see that

d m (d−1) m m(1− 1 ) S(Ax , p ) = p d = p d , a formula known to Hardy and Littlewood [26], [27], in connection with their work on Waring’s problem. Two useful observations come out of this example. First is the idea of a recursion formula for evaluating S(f, pm), an idea we elucidate in more generality in section 5, and second is the realization that the magnitude of such a sum can be as large m(1− 1 ) as p d .

3. Upper bounds on pure exponential sums Hua [30], [31], [32] established the following uniform upper bound on exponential sums: For any nonconstant polynomial f (mod p) of degree d and any prime power pm with m ≥ 1, m m(1− 1 ) (3.1) |S(f, p )| ≤ c1(d)p d , 3 with c1(d) = d . In view of the preceding example, the exponent here is best possible, but improvements on c1(d) were made by Chen [6], [7], Chalk [5], Ding [20], [21], Loh [58], [59], Lu [62], [63], Mit’kin [68], Ne˘caev[72], [73] and Ste˘ckin[89]. Ste˘ckinshowed that c1(d) could be taken as an absolute constant although he did not indicate how large it must be. In [14] we established that one may take c1(d) = 4.41, including the case p = 2. This value for c1(d) follows readily from (5.5).

Question 5. What is the best possible value for c1(d)? Is it possible to take c1(d) = 1+o(d) ? Ne˘caevand Topunov [74] determined the best possible constant to be 1.986 and 2.263 for polynomials of degree 3 and 4 respectively. Ne˘chaev[72], Chen [7] and the authors [14] obtained the optimal upper bound c1(d) = 1, under the additional assumption that 2d 1− 1 p > (d − 1) d−2 , the interval where Weil’s upper bound implies that |S(f, p)| ≤ p d . Using the multiplicative property of exponential sums one obtains corresponding upper bounds for S(f, q) with q arbitrary, but there still remains the basic question:

Question 6. What is the best possible constant c2(d) such that for an arbitrary modulus q, 1− 1 |S(f, q)| ≤ c2(d)q d , for any polynomial f that is nonconstant (as a function) (mod p) for each prime divisor p of q? EXPONENTIAL SUMS 5

d+O(d/ log d) Currently, the best values available for a general modulus are c2(d) = e due to 1.74d Ste˘ckin[89] and c2(d) = e , due to Qi and Ding [78]; see also Chen [6], [7], Lu [62], [63], Ne˘caev[72], [73], Qi and Ding [76], [77], and Zhang and Hong [96]. As noted by these authors the biggest obstacle to improving the value of c2(d) is the task of improving the Weil upper bound for (mod p) exponential sums when p is small relative to d.

4. On improving the Weil upper bound From (2.2) we have the basic upper bound of Weil for (mod p) exponential sums, √ (4.1) |S(f, p)| ≤ (d − 1) p. The bound is valid for any prime p and polynomial f of degree d over Z that is nonconstant (mod p), with the exception (p, d) = (2, 2). In one sense this bound is best possible. Mit’kin [67] proved that for any d and > 0 there exists an infinite family of pairs (p, λ) with p λ such that - √ |S(λxd, p)| ≥ (d − 1 − ) p. In this example d is fixed and the prime p grows arbitrarily large. The real interest is in improving the Weil upper bound when p is small relative to d, in particular when p < d2, where the bound is trivial. The only significant progress in this direction appears to be for the case of sparse polynomials, polynomials having relatively few nonzero terms in Pn di comparison with the degree. Mordell [70] showed that for the polynomial f = i=1 aix , with p - ai for all i and 1 ≤ d1 < d2 < ··· < dn, that 1 1− 1 (4.2) |S(f, p)| ≤ (d1d2 . . . dn(p − 1, d1, . . . , dn)) 2n p 2n . This is actually a slightly refined version of Mordell’s result as given by Shparlinski [83]. Thus for instance, if p - AB, then (4.3) |S(Axd + Bx, p)| ≤ d1/4p3/4, which is sharper than (4.1) for p < (d − 1)4/d. The case of Gauss sums S(Axd, p) has received much attention. Heath-Brown and Konyagin [29], sharpening earlier bounds of Konyagin and Shparlinski [52], Mullen and Shparlinski [71] and Shparlinski [84] established the following: For p - a, ( d5/8p5/8 (4.4) |S(Axd, p)| d3/8p3/4.

3 3 These bounds are sharper than Weil and nontrivial on the interval d 2 p d . Konyagin 4 + [51] has recently sharpened (4.4) to obtain nontrivial bounds for p > d 3 . Konyagin [50] and Konyagin and Shparlinski [52] give upper bounds of a much weaker type that are nontrivial for much smaller values of p relative to d. The reader is referred to the recent book of Konyagin and Shparlinski [53] for further discussion of this problem. A long-standing and very deep question is the following.

1 + Question 7. For any > 0 is there a constant C() such that |S(f, p)| ≤ C()(dp) 2 for any nonconstant polynomial f (mod p)? This type of upper bound is plausible if one believes that the d − 1 values ω in (2.2) are √ i randomly distributed on the circle of radius p. Montgomery, Vaughan and Wooley [69] gave a heuristic argument to suggest that for Gauss sums one has the bound |S(f, p)| (dp log(p))1/2; the bound they state is even more precise. As pointed out to the authors by Igor Shparlinski, results of Karatsuba [44, Theorem 4] and [45] show that for Gauss sums such a bound would be best possible; see also Levenstein [54]. In particular, it is √ not possible to obtain a uniform upper bound of the type |S(f, pm)| ≤ Cd1/2 p with C an absolute constant. (Mit’kin’s example above also prohibits such a bound.) A more modest question is the following. 6 TODD COCHRANE AND ZHIYONG ZHENG

Question 8. Obtain any nontrivial bound for S(f, p) when p < (d − 1)2, under the appropriate assumptions on f. For instance, under what restrictions on f does one have p √ |S(f, p)| ≤ 2 or |S(f, p)| ≤ p − p? It is important that certain restrictions be placed on f in view of examples such as S(xp − p−1 d x, p) = p and S(x 2 , p) = 1 + (p − 1)cos(2π/p). For the two term polynomial Ax + Bx progress can be made. Combining the bound in (4.3) with the work of Akulinichev [1] we are able to show that if p - AB and d 6≡ 1 (mod p − 1) then 7 (4.5) |S(Axd + Bx, p)| ≤ p. 8 5. Local upper bounds on pure exponential sums Chalk [5] established the following upper bound for any nonconstant polynomial f (mod p) and any m ≥ 2,

m X t/(M+1) m(1− 1 ) (5.1) |S(f, p )| ≤ d( να)p p M+1 , α∈A where να is the multiplicity of α as a zero of the critical point congruence and M = maxα∈A(να). He suggested that one may be able to eliminate the value d altogether from the right-hand side, and this was proven independently by Ding [21], Loh [59] and the authors [12], under the assumption p is odd and m ≥ t + 2. In [14] we went one step further and proved that for p odd and m ≥ t + 2 or p = 2 and m ≥ t + 3, m t m(1− 1 ) (5.2) |Sα(f, p )| ≤ λαp να+1 p να+1 , with λα = min{να, 3.06}, and thus m X t m(1− 1 ) X t (m−t)(1− 1 ) (5.3) |S(f, p )| ≤ ( λα)p M+1 p M+1 = ( λα)p p M+1 . α∈A α∈A 2d/(d−1) Moreover, if p > (d − 1) then we can take λα = 1. The second expression on the right-hand side gives a better reﬂection of the fact that the sum degenerates to a (mod pm−t) exponential sum in the manner of (2.9). P Question 9. Is it possible to replace α λα in (5.3) with an absolute constant? A related upper bound in which the value M is replaced by the maximum multiplicity 0 of the zeros of f (x) over C was considered by Smith [88], Loxton and Smith [60] and Loxton and Vaughan [61], the latter obtaining

m (δ+τ)/(M0+1) m(1− 1 ) (5.4) |S(f, p )| ≤ (d − 1)p p M0+1 , where M 0 is the maximum multiplicity of any of the complex zeros of f 0, τ = 0 if d < p, 0 0 0 τ = 1 if d ≥ p, and δ = ordp(D(f )), where D(f ) is the diﬀerent of f . In many cases one will have M 0 < M and so this upper bound will be stronger (as m grows large) than (5.3). Loxton and Vaughan give several examples for which the bound in (5.4) is essentially best possible. In application it is sometimes more useful to have upper bounds stated in terms of dp(f) −t 0 and dp(f1) the degrees of f and f1 read (mod p), where f1 = p f . In [14] Theorem 2.1 we proved that if p is odd and m ≥ t + 2 or p = 2 and m ≥ t + 3, and dp(f) ≥ 1, then t 1 m m(1− ) (5.5) |S(f, p )| ≤ 3.06 p dp(f1)+1 p dp(f1) . This essentially follows from (5.2). Under the same hypotheses it was shown in [13], inequality (5.1), that 1 m m(1− d (f) ) (5.6) |S(f, p )| ≤ dp(f1)p p , √ with an extra factor of 2 in case p = 2. If p > deg(f) then dp(f1) = dp(f) − 1 and so these two bounds are essentially of the same strength, but for smaller p it is possible to EXPONENTIAL SUMS 7

p d have dp(f1) > dp(f) in which case the latter bound is stronger; consider eg. f = x + px . The upper bound in (5.6) follows from the stronger bound given in (5.10). The proofs of all of the upper bounds stated in this section follow the same general inductive line of argument. For any polynomial h(X) over Z, let ordp(h(X)) denote the largest power of p dividing all of the coeﬃcients of h(X). For any critical point α deﬁne

(5.7) σ = σα = ordp(f(pY + α) − f(α)), and −σ (5.8) gα(Y ) = p (f(pY + α) − f(α)). Setting x = α + py with y running from 1 to pm−1 one readily obtains the following relationship for m ≥ σ.

m σ−1 m−σ (5.9) Recursion Relationship : Sα(f, p ) = epm (f(α))p S(gα, p ). m m−1 If m < σ then we just have Sα(f, p ) = epm (f(α))p . The upper bounds can then be obtained by induction on m. By successive applications of the recursion relationship one can often succeed in either explicitly evaluating S(f, pm) or at least expressing S(f, pm) in terms of (mod p) exponential sums. In either case one obtains S(f, pm) as a sum of √ complex numbers having moduli that are powers of p. As an example, suppose that we wish to evaluate S(f, p4), with f = 3x4 − 4x3 and p > 3. There are two critical points α = 0 of multiplicity 2 and α = 1 of multiplicity one. For α = 0 we have σ = 3, 3 4 2 3 gα(y) = y (3py − 4), and so Sα(f, p ) = p S(−4x , p). We are left with a cubic Gauss 4 2 4 sum (mod p). For α = 1, Sα(f, p ) = ep4 (−1)p . Thus altogether, S(f, p ) is a sum of two complex numbers of modulus p5/2 and one of modulus p2. Another local type of upper bound, in terms of the parameter σα, was established in [13], again using the recursion relationship above: For p odd m ≥ t + 2 and f nonconstant (mod p) we have for any critical point α,

m m(1− 1 ) (5.10) |Sα(f, p )| ≤ ναp σα .

Since σα ≤ να + 1 + t the upper bound here is often sharper than (5.2).

6. Pure exponential sums with rational function entries

Let f = f1/f2 be a nonconstant (mod p) rational function over Z with d = deg(f1) + ∗ deg(f2), and d = max{deg(f1), deg(f2)}. We deﬁne

pm m X S(f, p ) = epm (f1(x)f2(x)) x=1 p-f2(x) where the overline denotes multiplicative inverse (mod pm). The value t, the critical point congruence and the set of critical points are deﬁned identically as above, (2.3) and (2.4). In [13] we established that the equality in (2.8) holds for any rational f. Also, by the method of [14] it is not hard to show that the upper bound in (5.2) holds also for any nonconstant (mod p) rational f, and any m ≥ t + 2. In [13] we obtained the following analogue of the Hua upper bound: For any odd prime p and nonconstant (mod p) rational function f and m ≥ 2,

m m(1− 1 ) (6.1) |S(f, p )| ≤ dp d∗ . √ For p = 2 one needs an extra factor of 2 on the right-hand side. One cannot obtain the fraction 1/d∗ in the exponent directly from (5.3) since it is possible for the multiplicity of p k a critical point can be as large as d − 1; consider f = x /(1 + x ). However, over C the maximum multiplicity of any zero of f 0 is at most d∗ − 1, and so one might expect the bound in (6.1) in view of the result of Loxton and Vaughan (5.4). The upper bound here 8 TODD COCHRANE AND ZHIYONG ZHENG sharpens earlier results of Ismoilov [40] and Stepanov and Shparlinski [86]. It is deduced ∗ from (5.10) and the fact that σ ≤ dp(f).

Question 10. Is it possible to replace the value d on the right-hand side of (6.1) with an absolute constant?

Question 11. Does an upper bound of the type (5.4) hold for rational functions?

7. Mixed exponential sums Now let χ be a multiplicative character (mod pm) and let f(x), g(x) be rational functions over Z. We consider the general “mixed” or “hybrid” sum

pm m X S(χ, g, f, p ) = χ(g(x))epm (f(x)). x=1

It is understood that in the sum x runs only through values for which f(x) and g(x) are both deﬁned (mod pm) and g(x) is nonzero (mod p). For m = 1 it follows from the work of Weil [92] (see also Perel’muter [79]) that if the sum is nondegenerate (that is, either p k f 6= f1 − f1 for any f1 ∈ Fp[x] or g 6= g1 for any g1 ∈ Fp[x] where k is the order of χ) then √ (7.1) |S(χ, g, f, p)| ≤ (n1 + n2 − 2 + deg(f)∞) p, where n1 is the number of poles or zeros of g, n2 is the number of poles of f and (f)∞ is Pn2 the divisor (f)∞ = i=1 miPi with mi the multiplicity of the pole Pi. For m ≥ 2 our strategy for estimating the mixed exponential sum is to convert it into a pure exponential sum. Suppose that p is odd and let a be a primitive root (mod pm). In order to evaluate the mixed exponential sum we work over the ring of p-adic integers Zp and make use of the p-adic logarithm

∞ X (−1)i−1(px)i log(1 + px) = , i i=1

p−1 which takes on p-adic integer values for x ∈ Zp. Deﬁne r by a = 1+rp and let R be the p-adic integer R = p−1 log(1 + rp). Let c be the unique integer with 1 ≤ c ≤ pm−1(p − 1) such that for any integer k,

2πick χ(ak) = e pm−1(p−1) .

We extend the additive character epm (·) to Zp by setting for x ∈ Zp, epm (x) = epm (˜x) m m wherex ˜ is the image of x in Zp/(p ) ' Z/(p ). The key to untwisting the mixed exponential sum is to observe that the multiplicative character χ acts on the subgroup of residues congruent to 1 (mod p) in the following manner; see [9, Lemma 2.1]. For any p-adic integer y,

(7.2) χ(1 + py) = epm (Rc log(1 + py)).

In particular,

m/2 (7.3) χ(1 + p y) = epm/2 (Rcy).

To illustrate the argument, suppose that m is even. Writing x = u + pm/2v with u and v running from 1 to pm/2 one obtains from (7.3) EXPONENTIAL SUMS 9

m X X m/2 m/2 S(χ, g, f, p ) = χ(g(u + p v))epm (f(u + p v)) u v X X 0 m/2 0 m/2 = χ(g(u) + g (u)p v)epm (f(u) + f (u)p v) u v X X 0 0 = χ(g(u))epm (f(u)) epm/2 (Rcg(u)g (u)v + f (u)v) u v m/2 X = p χ(g(u))epm (f(u)). cg0(u)+Rgf 0(u)≡0 (mod pm/2) This leads us to deﬁne a critical point congruence (7.4) C(x) := p−t Rg(x)f 0(x) + cg0(x) ≡ 0 (mod p),

0 0 where t = ordp(Rgf + cg ), together with a set of critical points

(7.5) A = {α ∈ Fp : C(α) ≡ 0 (mod p) and g(α) 6≡ 0 (mod p)}. We again have the basic decomposition,

m X m S(χ, g, f, p ) = Sα(χ, g, f, p ), α∈A for p odd and m ≥ t + 2 or p = 2 and m ≥ t + 3. Also, if p is odd, m ≥ t + 2 and α is a critical point of multiplicity one then  ∗ ∗ m+t m 2 m χ(g(α ))ep (f(α ))p , m − t even; (7.6) Sα(χ, g, f, p ) = ∗ ∗ A m+t−1 m α 2 χ(g(α ))ep (f(α )) p Gpp , m − t odd; where α∗ is the unique lifting of α to a solution of the congruence

C(x) ≡ 0 (mod p[(m−t+1)2]),

0 and Aα ≡ 2r(C/g) (α) (mod p). The details of this derivation are worked out in [9], but the essence of the argument is as given above. If each critical point has multiplicity one then for p odd and m ≥ t + 2 we obtain the formula  ∗ ∗ m+t P m 2 m  α∈A χ(g(α ))ep (f(α ))p , if m − t even; (7.7) S(χ, g, f, p ) = ∗ ∗ A m+t−1 P m α 2  α∈A χ(g(α ))ep (f(α )) p Gpp , if m − t odd;

m+t a sum of complex numbers of moduli p 2 . Suppose now that f and g are polynomials. It is clear that there are at most deg(f) + deg(g)−1 critical points, but we can be even more precise. Let n be the number of distinct roots of g over Fp. Then writing the critical point congruence in the manner g0 (7.8) p−t(Rf 0 + c ) ≡ 0 (mod p), g we see that the number of critical points is at most deg(f) + n − 1. Interestingly, this is the same value that appears in the corresponding (mod p) exponential sum. To be precise, if f, g are polynomials such that p deg(f) and (ord(χ), deg(g)) = 1 then there - √ exist complex numbers ωi, 1 ≤ i ≤ deg(f) + n − 1, of moduli p such that

S(χ, g, f, p) = ω1 + ω2 + ··· + ωdeg(f)+n−1; see [82] or [90]. We again ask Questions 2 and 3 in this more general setting. 10 TODD COCHRANE AND ZHIYONG ZHENG

8. Upper bounds on mixed exponential sums For critical points of multiplicity greater than one, we obtain [9] an analogue of the recursion formula stated for pure exponential sums. Suppose that f and g are rational functions and that α is a critical point of multiplicity ν. Let (8.1) Fα(Y ) := cR log g(α)g(α + pY ) + f(α + pY ) − f(α).

Fα(Y ) may be expanded into a formal power series of the type ∞ X j (8.2) Fα(Y ) = aj Y , j=1 with p-adic integer coeﬃcients aj . Deﬁne

(8.3) σ := ordp(Fα(Y )) = min(ordp(aj )), j≥1

−σ (8.4) Gα(Y ) := p Fα(Y ).

Note, for the case of pure exponential sums Gα(Y ) and σ coincide with the definitions given earlier in (5.8), (5.7). We have for m ≥ σ the m σ−1 m−σ (8.5) Conversion F ormula : Sα(χ, g, f, p ) = p χ(g(α))epm (f(α))S(Gα, p ), m−σ Ppm−σ where S(Gα, p ) = y=1 epm−σ (Gα(y)). The function Gα, defined apriori as an infinite series with p-adic integer coefficients, may be viewed as a polynomial over Z in m−σ the exponential sum S(Gα, p ), since its coefficients are p-adic integers and the high order coefficients all vanish modulo pm−σ. Thus we have succeeded in reducing the mixed m−σ exponential sum to the pure exponential sum S(Gα, p ). We proceed now to obtain a relationship between Gα and the critical point function C defined in (7.4). Note, g0(α + pY ) C(α + pY ) (8.6) F 0 (Y ) = cRp + pf 0(α + pY ) = pt+1R . α g(α + pY ) g(α + pY ) Develop RC/g into a Taylor expansion about α, ∞ C(X) X j (8.7) R = c (X − α) , g(X) j j=0 with p-adic integer coefficients cj , and note that since α is a zero of C of multiplicity ν

(8.8) ordp(cj ) > 0 for 0 ≤ j < ν and ordp(cν ) = 0. It follows from (8.6) that ∞ j+1 t+1 X j Y (8.9) F (Y ) = p c p , α j j + 1 j=0 and that ∞ ∞ −σ −σ X j t−σ X cj−1 j j (8.10) G (Y ) = p F (Y ) = p a Y = p p Y . α α j j j=1 j=1 Using local upper bounds of the type (5.5) and (5.6) we are then able to establish [9] that (5.2) and (5.3) hold identically for any mixed exponential sum under the assumption m ≥ t + 2 (p odd). In particular, if p is odd, f, g are rational functions over Z and m ≥ t + 2 then m X t m(1− 1 ) (8.11) |S(χ, g, f, p )| ≤ ( λα)p M+1 p M+1 , α∈A where M is the maximum multiplicity of any critical point. A similar upper bound holds for p = 2; see [9] section 7. EXPONENTIAL SUMS 11

If χ is not a primitive character (mod pm) and f 0 is not identically zero (mod p) then M ≤ d − 1, where d = deg(f1) + deg(f2) is the total degree of f = f1/f2. We recover a Hua-type upper bound, m t m(1− 1 ) (8.12) |S(χ, g, f, p )| ≤ 3.06 p d p d , for p odd and m ≥ t + 2. Suppose now that χ is a primitive character. Writing f = f1/f2, g = g1/g2, d = deg(f1) + deg(f2) and letting n being the number of distinct zeros of g1g2 over Fp, it follows from (7.8) that the maximum multiplicity M of any critical point satisﬁes,

(8.13) M ≤ max{deg(f) + n − 1, 2deg(f2) + n − 1}.

If deg(f2) ≤ deg(f1) then we deduce from (8.11) the upper bound m X t m(1− 1 ) (8.14) |S(χ, g, f, p )| ≤ ( λα)p d+n p d+n , α∈A for p odd and m ≥ t + 2. Question 12. Can the upper bound on M in (8.13) be improved for certain values of deg(f1), deg(f2) and n?

If f, g are polynomials of degrees d1, d2, we can state a more precise uniform upper bound. If p is odd, m ≥ 2 and S(χ, g, f, pm) does not degenerate to a sum of smaller modulus then we have [9, Corollary 1.1], m m 1− 1 (8.15) |S(χ, g, f, p )| ≤ 4.41 p d1+d2 . For p = 2 the same holds with constant 8.82 in place of 4.41. The sum degenerates to one of smaller modulus if dp(f) = 0 and either χ is not primitive or dp(g) = 0. This bound includes as a special case the upper bound of Hua for pure exponential sums, and the upper bound in section 11 for character sums. Questions 5 and 6 may again be asked for this more general sum. Perhaps the most important question to ask in regard to mixed exponential sums, especially in view of the important role pure exponential sums have played in number theory and elsewhere is the following. Question 13. Are there any applications of mixed exponential sums and of the conversion formula and estimates stated in this section? In the next ﬁve sections we consider a number of special types of pure and mixed exponential sums.

9. The twisted exponential sum S(χ, x, f, pm). The author’s exploration of mixed exponential sums started with the consideration of m the twisted exponential sum S(χ, x, f, p ) with f a polynomial of degree d over Z. It had been conjectured by E. Bombieri, M. Liu and W.M. Schmidt, that an upper bound analagous to that of Hua should be available for this sum. One might even expect a stronger uniform upper bound than Hua due to extra cancellation coming from the twisted terms, but just the opposite occurs. If f and χ are chosen so that the critical point congruence p−t(Rxf 0(x) + c) ≡ 0 (mod p), has a single zero of multiplicity d then the upper bound of Hua can fail to hold. In [12], m(1− 1 ) Example 9.2, we give an example of a polynomial f with S(χ, x, f, pm) = p d+1 for inﬁnitely many values of m. From (8.15) we have the uniform upper bound

pm X m(1− 1 ) (9.1) | χ(x)epm (f(x))| ≤ 4.41 p d+1 , x=1 12 TODD COCHRANE AND ZHIYONG ZHENG valid for any odd p, any nonconstant (mod p) polynomial f and any m ≥ 1. Again, for p = 2 one needs an extra factor of 2 on the right-hand side. In most cases there will not be a critical point of multiplicity d and so sharper bounds are available. Question 14. Can an analogue of the upper bound of Loxton and Vaughan [61] be stated for the twisted exponential sum S(χ, x, f, pm)?

For rational functions f = f1/f2 with dp(f1) + dp(f2) ≥ 1 and m ≥ 2 we established in [13] that

pm X m(1− 1 ) (9.2) | χ(x)epm (f(x))| ≤ 4Dp D+1 , x=1 where D = max(deg(f), 2deg(f2)). Also, in Example 6.1 of [13] we give examples where m m(1− 1 ) deg(f1) = deg(f2) and |S(χ, x, f, p )| = p D+1 for inﬁnitely many m.

10. Gauss sums For the Gauss sum S(χ, x, x, pm) the critical point congruence is Rx+c ≡ 0 (mod p). If χ is imprimitive, that is p|c, then there is no critical point, and the sum is zero for m ≥ 2. If χ is primitive, then there is a single critical point α ≡ −cR (mod p) of multiplicity one, and by (7.7)

pm ( ∗ ∗ m/2 X χ(α )epm (α )p , if m is even; m (10.1) χ(x)ep (x) = ∗ ∗ −2Rc m−1 χ(α )e m (α ) G p 2 , if m > 1 is odd, x=1 p p p where α∗ ≡ −cR (mod p[(m+1)/2]). A variation of this formula was obtained by Odoni [75] and Mauclaire [65], [66]; see also the book by Berndt, Evans and Williams [2].

11. Two term exponential sums The sum S(Axd + Bx, pm) arises in the application of the circle method to Waring’s problem; see [91]. For this sum Hua [31] obtained the upper bound

pm X d m/2 m | epm (Ax + Bx)| ≤ c(d)p (p ,B), x=1 for d ≥ 2 and p - A. Loxton and Smith [60], Smith [88], Dabrowski and Fisher [17] and Ye [95] made further reﬁnements. In [15], as an application of the method presented here, the authors obtained pm X d m/2 m 1/2 (11.1) | epm (Ax + Bx)| ≤ dp (p ,B) . x=1 A more precise version of our upper bound may be stated in terms of the parameters β, δ and h, deﬁned by pβ kd, pδkB, phk(d − 1). For m ≥ 2 we have

pm X d δ/2 h/2 1 min(1,β) m/2 (11.2) | epm (Ax + Bx)| ≤ (d − 1, p − 1)p p p 2 p . x=1 This upper bound is nearly best possible but there is still room for improvement.

1 min(1,β) Question 15. Can the term p 2 be eliminated from the right-hand side? Can the value (d − 1, p − 1) be improved? EXPONENTIAL SUMS 13

For the mixed exponential sum S(χ, x, Axd + Bx, pm) we established in [16] the upper bound

pm X d 2m/3 (11.3) | χ(x)epm (Ax + Bx)| ≤ 2dp , x=1 and showed that the exponent 2m/3 is best possible in general. This upper bound is valid for any prime p, m ≥ 1, d ≥ 2 and p - (A, B). If χ is a character with conductor p (eg. the principal character or Legendre symbol), p - (A, B), d ≥ 2, m ≥ 1 then we obtain [16] the stronger bound

pm X d m/2 (11.4) | χ(x)epm (Ax + Bx)| ≤ dp . x=1 A more precise statement analogous (11.2) is stated in Theorem 1.3 of [16]. It is of interest to note that for the mixed exponential sum in (11.4) we are able to eliminate the factor (pm,B)1/2, required for the upper bound in (11.1). √ 1 + Question 16. Can the value d in (11.4) be replaced by O( d) or O(d 2 )?

12. Sparse Polynomials

d1 d2 dn Let f be a sparse polynomial of the type f(x) = a1x + a2x + ··· + anx , with 1 ≤ d1 < d2 < ··· < dn = d and the ai nonzero integers. For the case of (mod p) exponential sums we already noted in (4.2) that an upper bound exists that is sometimes sharper than the Weil bound. For m ≥ 2 upper bounds exist that are sharper than the Hua upper bound provided that we restrict x to nonzero residues (mod p). If p > d then one can show that the maximum possible multiplicity of any nonzero critical point m associated with the sum S(f, p ) is n − 1. Thus by (5.3) it follows that if dp(f) ≥ 1, then

pm X 3/2 m(1− 1 ) (12.1) | epm (f(x))| ≤ d p n . x=1 p-x If p ≤ d then there may exist zeros of larger multiplicity (mod p). However, over C, the multiplicity will still be at most n − 1. Thus one can use the upper bound of Loxton and Vaughan (5.4) to obtain an upper bound with the same exponent on p (but diﬀerent constant). General upper bounds of the same order of magnitude have been obtained by Loh [59] and Shparlinski [85]. Shparlinski dealt also with Laurent polynomials and showed a connection between these sums and exponential sums with linear recurrence sequences.

13. Character Sums By a character sum we mean a sum of the type

pm m X S(χ, g, 0, p ) = χ(g(x)), x=1 with g = g1/g2 a nonconstant rational function over Z and χ a multiplicative character (mod pm). We may assume that χ is a primitive character for otherwise the sum degenerates to one of smaller modulus. Such sums were ﬁrst studied by Ismoilov [33], [34], [35], [36], [37], [38], [39], [41] and then by Liu [55], [56], [57]. The critical point congruence for the sum is p−tg0(x) ≡ 0 (mod p). 14 TODD COCHRANE AND ZHIYONG ZHENG

It is plain that the maximum possible multiplicity of any critical point is d − 1 where d is the total degree of g and so we obtain from (8.11), for p odd and m ≥ t + 2,

pm X m(1− 1 ) (13.1) | χ(g(x))| ≤ d p d . x=1 Ismoilov and Liu stated bounds of comparable strength, some being more precise. Ismoilov noted that the exponent in (13.1) is best possible in general. Indeed, if g = Axd + B, m m(1− 1 ) d ≥ 2, p - ABd and d|m then for any primitive character χ, |S(χ, g, 0, p )| = p d . Ismoilov [41] also indicated that in certain cases the value d in the exponent of (13.1) may ∗ be replaced by d := max(deg(g1), deg(g2)). In [10] Cochrane, Liu and Zheng established m the more general result that if dp(g) ≥ 1 and χ is a primitive character (mod p ) then for any m ≥ 2 we have

m m(1− 1 ) (13.2) |S(χ, g, p )| ≤ (d − 1)p d∗ , with an extra factor of 2 on the right-hand side in case p = 2. Question 17. Can the constant (d − 1) in (13.2) be replaced by an absolute constant? For the case of polynomial g the answer is yes. From (8.15) we have the upper bound

pm X m(1− 1 ) (13.3) | χ(g(x))| ≤ 4.41 p d , x=1 m for any odd p with dp(g) ≥ 1, m ≥ 2 and any primitive character χ (mod p ). The analogues of Questions 6,7 and 8 for character sums are all important here. For a general modulus q, Liu [56], [57] obtained the upper bound q X 1.8d 1− 1 | χ(g(x))| ≤ e q d , x=1 for any rational g that is nonconstant (mod p) for each p|q, and any primitive character χ (mod q). The authors are not aware of any progress that has been made towards sharpening the basic upper bound of Weil for prime modulus, when d is large relative to p.

14. Kloosterman and Salie´ sums Kloosterman and Saliésums of the type S(Ax + Bx−1, pm) and S(χ, x, Ax + Bx−1, pm), respectively, with p - AB, occur as Fourier coefficients of certain automorphic forms; see Iwaniec [43], p. 78. It was Kloosterman [48] who first noted this connection. The Kloosterman sum for m ≥ 2 was evaluated by Salié[80], and later by Whiteman [93], √ Estermann [23], Carlitz [4] and Williams [94]. For m = 1 we have the upper bound 2 p due to Weil [92], but the evaluation is unknown. For the case of the quadratic character χ, odd p and m ≥ 1, the sum S(χ, x, Ax + Bx−1, pm) was evaluated by Iwaniec [42], [43] p. 68 and Sarnak [81], p. 90. For p = 2 an evaluation for quadratic χ was given by Dedeo [18], chap. 4. Duke, Friedlander and Iwaniec [22] have found applications of such estimates in their work on bilinear forms with Kloosterman fractions. Suppose that p is odd. The critical point congruence for such a sum is RAx2 + cx − RB ≡ 0 (mod p). Let ∆ = c2 + 4R2AB. It follows that if ∆ is a quadratic nonresidue (mod p) then the sum is zero while if ∆ is a quadratic residue (mod p) then there are two critical points of EXPONENTIAL SUMS 15 multiplicity one, and we obtain the classical formula for the Kloosterman and Saliésum. If ∆ ≡ 0 (mod p) then there is a single a critical point of multiplicity two. In any case, m p ( m/2 X −1 2p , p - ∆; m | χ(x)ep (Ax + Bx )| ≤ 2m/3 x=1 2p , p|∆. A more detailed discussion may be found in [13], including the case p = 2. A multidimensional (or hyper) Kloosterman sum is defined by

pm pm m X X Kn(A, p ) := ··· epm (x1 + x2 + ··· + xn + A(x1x2 . . . xn))

x1=1 xn=1 with p - A. Smith [87] established that for p odd and m ≥ 1, m nm/2 |Kn(A, p )| ≤ (n + 1)p , the case m = 1 being due to Deligne [19]. Dabrowski and Fisher [17, Example 1.17] sharpened the estimate of Smith for certain values of n, to which Ye [95] made an application. In [11] the authors, with M.-C. Liu, made a further sharpening to obtain

m 1 min(γ,m−2) mn/2 (14.1) |Kn(A, p )| ≤ (n + 1, p − 1)p 2 p , for any odd prime p and m ≥ 2, where pγ k(n + 1). For p = 2 the same bound is obtained with an extra factor of 2 on the right-hand side. Katz [46] studied general twisted hyper-Kloosterman sums over ﬁnite ﬁelds. Evans [25], [24] has recently made evaluations and estimates for twisted hyper-Kloosterman sums modulo pm with m ≥ 2, and for Kloosterman sums over rings of algebraic integers. Question 18. Can an upper bound of the type (14.1) be established for twisted hyper- Kloosterman sums?

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Department of Mathematics, Kansas State University, Manhattan, KS 66506 E-mail address: [email protected]

Department of Mathematics, Tsinghua University, Beijing 100084, P.R. China E-mail address: [email protected]