n-dimensional norm Kloosterman sums over the ring of the Gaussian numbers investigate. Nontrivial estimates of these sums were obtained

Savastru, O.

Varbanets, S.

Algebra and Discrete Mathematics

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Algebra and Discrete Mathematics RESEARCH ARTICLEVolume 11 (2011). Number 2. pp. 82 – 91c© Journal “Algebra and Discrete Mathematics”Norm Kloosterman sums over Z[i]O. Savastru and S. VarbanetsCommunicated by V. V. KirichenkoAbstract. n-dimensional norm Kloosterman sums over thering of the Gaussian numbers investigate. Nontrivial estimates ofthese sums were obtained.IntroductionThe classical Kloosterman sums K(a, b ; q) and their n-dimensional ana-logue Kn(a0, a1, . . . , an ; q) are important ingradients in problems con-tained with a distribution of arithmetic functions on the arithmetic pro-gressions (see, [5], [6] and etc.). The sums of the Kloosterman sums studiedin the works N.V. Kuznetsov [12], H. Iwaniec and I.-M. Deshouillers [7]and etc., have allowed to improve the error terms in additive problemsof number theory. The same problems arise at the decision of the asymp-totic tasks over the ring of integer numbers of the ﬁnite expansions ofthe rational ﬁeld. For example, U. Zanbyrbaeva [11] obtained a nontivialestimate of error term in the problem of divisors of the Gaussian integersin an arithmetical progressions (see. also [9]), and in 2003 Bruggeman andY. Motohashi [2] obtained the lower estimate for fourth moment of thezeta–function Hecke of the Gaussian ﬁeld Q[i].In 2008 was considered the norm Kloosterman sum over the ring of theGaussian integers Z[i] (see. [8]). It permitted to ﬁnd non–trivial asymp-totic formula for the divisor function τ(α) with norm in an arithmeticalprogression.In present paper we study n–dimensional norm Kloosterman sumsover the ring of the Gaussian integers.2000 Mathematics Subject Classification: 11L05.Key words and phrases: Kloosterman sum, Gaussian numbers.O. Savastru, S. Varbanets 831. PreliminariesIn this section we ﬁx some notations and gather some necessary facts thatwill be used in the sequel.In the following, greek letters α, β, . . . we denote the Gaussian integers; pdenotes a prime rational number;Sp(α) is a trace α from Q[i] in Q, i.e. Sp(α) = 2ℜα;N(α) is a norm α, N(α) = αα = |α|2;Zq (respectively,Z∗q) is complete (respectively, reduced) system of residuesmodulo q in Z;R(q) (respectively,R∗(q)) is complete (respectively, reduced) system ofresidues modulo q in Z[i];notation∑S(C)means, that summation passes under the condition C, whichwe describe separately;(a, b) means the greatest common divisor a and b in Z (or in Z[i]);e2piixq := eq(x), x ∈ R.Let q = pa, a ∈ N. By km denote the ﬁeld with qm elements,m = 1, 2, . . . ; k1 := k. Let X is N -dimensional vector ∈ kN . Denote byVr is an algebraic variety deﬁned by polynomial f(X) ∈ k[X], i.e.Vr ={X ∈ kNr | f(X) = 0}.Let yet Tr(x), x ∈ kn, be the trace transformation kn → k deﬁned bythe Frobenius automorphism:Tr(x) := x+ xq + . . .+ xqn−1, x ∈ kn.Obviously that Tr(x) ∈ k. Put for a ∈ k∗, k∗ = k \ {0},Sr(f) =∑X∈Vrep (a Tr (f(X))) . (1)Introduce the functionζ(V, t, f) := exp(∞∑r=1Sr(f)rtr), (2)which we call the zeta-function of an algebraic variety V over the ﬁniteﬁeld k.B.Dwork [4] proved, that ζ(V, t, f) is a rational function h(t)g(t) , moreoverthe set of roots Ω = {ω−1} of the polynomials h(t), g(t) describes thefollowing lemmas:84 Norm Kloosterman sums over Z[i]Lemma 1 (Deligne [3]). For numbers ω−1j ∈ Ω we have|ωj | = qmj2 , mj are nonnegative numbers.Moreover, for every ω−1j ∈ Ω all complex conjugate have equal modules.(The numbers call characteristic roots, and numbers mj call the weightsof ωj , ω−1j ∈ Ω).Lemma 2. The cardinality of Ω does not exceed (4D + 5)2n+1, whereD := max(1 + degV, degf), i.e. depends only on n and f .Taking a logarithm from LHS and RHS of (2) and comparing coeﬃ-cients for the same powers t we obtainSr := Sr(V, f) =∑ih(ω−1i )=0(ωi)r −∑jg(ω−1j )=0(ωj)r. (3)Lemma 3. Let q be the number of elements of finite field k. Then for thecoefficients Sr associated with the zeta-function of algebraic variety overk the following assertions:(i) |Sr| = O(qrt2 ) if |Sr| = O(qrθ) for some θ > 0, where t = [2θ],(ii) if t ∈ N and |Sr| 6 cqrt2 for all sufficiently large r, then characteristicroots of zeta-function have the weight m 6 t, and the number ofroots does not exceed c2,hold.Lemmas 2 and 3 were proved by E. Bombieri [1].Deﬁne the algebraic variety V over k by the polynomialx1 . . . xN − a, a ∈ k∗, and the function f we deﬁne as f = x1 + . . .+ xN .In this special case P. Deligne [3] proved that ζ(V, t, f) has exactly Ncharacteristic roots of the weight (N − 1), and hence|Sr(V, f)| = |Sr| 6 Nqr(N−1)2 . (4)Now we deﬁne the N -dimensional Kloosterman sum.Let q > 1 is a positive integer. Then h ∈ N, (h, q) = 1, calls a normresidue modulo q, if there exist α ∈ Z, such that N(α) ≡ h(mod q).Since for (h, p) = 1, p > 2 is a prime,Jpn(h) :={x, y ∈ Zpn : x2 + y2 ≡ h (mod pn)}= pn(1−(−1)p−12p)O. Savastru, S. Varbanets 85we conclude that every h ∈ Z∗q is a norm residue modulo q if (q, 2) = 1.For the Gaussian integers α0, α1, . . . , αN deﬁne the N -dimensionalKloosterman sum for (h, q) = 1:K˜N (α0, α1, . . . , αN ; q, h) :=∑S(C)eq (ℜ(α0x0 + . . .+ αNxN )) , (5)гдеC : xi ∈ R(q), i = 0, 1, . . . , N ; N(x0x1 . . . xN ) ≡ h (mod q).For q = q1q2, (q1, q2) = 1, we haveK˜N (α0, α1, . . . , αN ; q, h) == K˜N (α0q2, . . . , αNq2; q1, h)K˜N (α0q1, . . . , αNq1; q2, h).(6)Therefore in sequel we consider only case q = pm, p is a prime, m ∈ N.From the deﬁnition of norm sum we have the trivial estimate|K˜N (α0, α1, . . . , αN ; pm, h)| 6 (ϕ(pm))2N+1, (7)where ϕ(·) is the Euler function.From an estimate of the n-dimensional Kloosterman sum over Z[i](see. [10]) we infer|K˜N (α0, α1, . . . , αN ; pm, h)| 6 ǫp ·N · (ϕ(pm))2N ,whereǫp(N) =N, if m is even number;N(N−1)2 · p2−N2 , if m is odd, p 6= 2, 3;4N, if m is odd, p = 2;3N, if m is odd, p = 3.From main theorem which will prove below we obtain "the rooted estimate"for K˜N (α0, α1, . . . , αN ; pm, h) (i.e. an estimate is the square root out oftrivial estimate) for the case (α0 . . . αN , p) = 1).2. Main resultLet γ ∈ Z[i]. Denote mγ = max{m : pm|γ}.Theorem 1. Let h is a norm residue modulo p. ThenK˜N (α0, α1, . . . , αN ; pm, h) 6 2(4N − 1)p2N(m−n)I(α1, . . . , αN ; pm), (8)86 Norm Kloosterman sums over Z[i]where I(α1, . . . , αN ; pm) is the number of solutions of the system of con-gruences (23);moreover I(α1, . . . , αN ; pm) 6 (4N − 1)pN(2n−m),if mα1 = . . . = mαN = 0.(Here n =[m+12], [x] denotes the biggest integer 6 x).Proof. First let m = 1. The case mα0 = . . . = mαN = 1 is trivial, and sosuppose αi is coprime with p for some number i. Puttingαj = aj + ibj , j = 0, 1, . . . , N , infer (a0, . . . , aN , b0, . . . , bN , p) = 1.For p ≡ 1(mod 4) we haveK˜n(α0, . . . , αN ; p, h) =∑S(C)ep(a0x0+. . .+aNxN−b0y0−. . .−bNyN ), (9)whereC : {x0, . . . , xN , y0, . . . , yN ∈ {0, 1, . . . , p−1};N∏j=0(x2j+y2j ) ≡ h (mod p)}.Let ε0 is solution of the congruence x2 ≡ −1(mod p).Putuj = xj + ε0yj , vj = xj − ε0yj , j = 0, 1, . . . , N. (10)By (h, p) = 1 and ujvj ≡ x2j + y2j 6≡ 0(mod p), j = 0, 1, . . . , N we obtainthat uj , vj ∈ Z∗p.Therefore (by (10)):K˜n(α0, . . . , αN ; p, h) =∑S(C)ep(N∑j=0Ajuj +N∑j=0Bjvj),C : {uj , vj ∈ Z∗p, j = 0, 1, . . . , N :N∏j=0ujvj ≡ h (mod p)}.(11)Let among of α0, . . . , αN have exactly l nulls modulo p elements (forexample, α0 = . . . = αl−1 = 0, αl+1 . . . αN 6≡ 0(mod p).Then from (11) inferK˜n(α0, . . . , αN ; p, h) =p−1∑u0=1. . .p−1∑ul−1=1∑S(C)ep(N∑j=l(Ajuj +Bjvj)),C : {uj , vj ∈ Z∗p, j = l, . . . , N ;N∏j=lujvj ≡ hl−1∏j=0(ujvj)−1(mod p)}.(12)Take into account that for j = l, . . . , N .Aj ≡12(aj − ε−10 bj), Bj ≡12(aj + ε−10 bj), ε0ε−10 ≡ 1 (mod p);O. Savastru, S. Varbanets 87AjBj ≡14(a2j + b2j ) ≡14N(αj) (mod p).Hence (Aj , p) = (Bj , p) = 1.Now, the application of the P. Deligne estimate [3] (see, formula (4), theabove) gives|K˜N (α0, α1, . . . , αN ; p, h)| 6 (N−l+1)p2Nlp2(N−l)−12 = (N−l+1)pN+l+12 .(13)In particular, for l = 0 we obtain "the rooted estimate" (N + 1)p2N+12 .Now consider the case p ≡ 3(mod 4).Let α0, . . . , αl−1 ≡ 0(mod p), αl . . . αN 6≡ 0(mod p).The complete system of residues modulo p in Z[i] is ﬁeld R(p) from p2elements.Again we consider the algebraic variety V over k = Zp, generated by thepolynomialx0x1 . . . xN − h, h ∈ k∗and let f(x0, . . . , xN ) = α0x0 + . . .+ αNxN .For p ≡ 3(mod 4) we have(u+ vi)p ≡ up − ivp ≡ u− iv (mod p), u, v ∈ Z,and, hence, in the ﬁeld R(p):Sp (u+ iv) ≡ Tr (u+ iv) (mod p), (14)where the function Sp(·) (respectively, Tr(·)) we consider as a function oftrace from the quadratic expansion Q[i] (respectively, from k2 = R(p)) inQ (respectively, in k = Zp).Hence, denoting by l the number of αi, congruented with 0 modulo p, wecan estimate S2(V, f) by formula (4). And so we haveS2(V, f) 6 (N + 1− l)p2(2N+2−2l−1)2 p4l = (N + 1− l)p2N+2l+1. (15)Now from (3) follows thatS21(V, f) ∼ NS2(V, f). (16)Thereby in the case p ≡ 3(mod 4) the assertion of theorem proved also.Since, the case p = 2 is trivial we obtain the proof of theorem for m = 1.Put further m > 1.Without loss of generality, we can suppose that(pmα0 , pmα1 , . . . , pmαN , pm) = 1. Consequently, though one from numbersaj , j = 0, 1, . . . , N is coprime whith p. Let(α0, p) = 1. Hence, (a0, b0, p) =88 Norm Kloosterman sums over Z[i]1, where α0 = a0 + ib0.Denote ξj = xj + iyj , j = 0, 1, . . . , N . We haveK˜N (α0, . . . , αN ; pm, h) == 1pm∑ξj∈R(pm)j=0,...,Npm−1∑k=0epm(k(N(ξ0 . . . ξN )− h) +N∑j=0ℜ(αjξj))== 1pm∑S(C)epm(kN∏j=0(x2j + y2j )− h) +N∑j=0ajxj −N∑j=0bjyj),(17)C : {k ∈ Zpm , xj , yj ∈ Zpm , j = 0, 1, . . . , N}.We can suppose that ξj ∈ R∗(pm), j = 1, . . . , N , (else, the summationover k gives zero).Next, for every k ≡ 0(mod p) the summation over x0, y0 (by(a0, b0, p) = 1) gives zero.Hence, from (14) obtainK˜N (α0, . . . , αN ; pm, h) = 1pm∑k∈Z∗pm∑ξj∈R∗(pm)j=1,...,Nepm(−kh)××∑ξ0∈R(pm)epm(kN(ξ0)N∏j=1N(ξj) + ℜ(α0ξ0) +N∑j=1ℜ(αjξj))== 1pm∑k∈Z∗pmepm(−kh)××∑ξj∈R∗(pm)j=1,...,N∑x0,y0∈Zpmepm(kN∏j=1N(ξj)(x20 + y20) + a0x0 − b0y0)==∑k∈Z∗pmepm(−kh)∑ξj∈R∗(pm)j=1,...,Nepm(ℜ(αjξj)− 4′k′N∏j=1N(ξj)′(a20 + b20)),(18)where 4·4′ ≡ 1(mod pm), k ·k′ ≡ 1(mod pm),N(ξj)·N(ξj)′ ≡ 1(mod pm),j = 1, . . . , N .(We estimated the sums over x0, y0 as the classical Gaussian sums).Let n := [m+12 ]. Put for j = 1, 2, . . . , N :ξj = ηj + pnζj ηj ∈ R∗(pn), ζj ∈ R(pm−n). (19)ThenN(ξj)′ = N(ηj)′(1− 2pn(xjuj + yjvj)N(ζη)′), (20)if we setηj = xj + iyj , ζj = uj + ivj , xj , yj ∈ Zpm , uj , vj ∈ Zpm−n (21)O. Savastru, S. Varbanets 89And so, from (18) we ﬁnd|K˜(α0, . . . , αN ; pm, h)| == |∑k∈Z∗pmepm(−kh)N∑j=1∑xj ,yj∈Zpn(x2j+y2j ,p)=1epm(ℜ(αjηj))∑uj ,vj∈Zpm−nepm(G+ pnH)|,(22)whereG = 4′k′(a20 + b20)N∏j=1(x2j + y2j )′ := 4′k′(a20 + b20)D.H =N∑j=1{−2′k′(a20 + b20) ·D · (xjuj + yjvj) + ajuj − bjvj}.Note that now the summation over uj , vj gives zero if the system ofcongruences {aj − 2D2N(ηj)′xj ≡ 0 (mod pm−n),bj + 2D2N(ηj)′yj ≡ 0 (mod pm−n),(23)disturbs though for one j, j = 1, 2, . . . , N . This system is equivalent tothe systemajyj + bjxj ≡ 0 (mod pm−n),2′k′N∏j=1((x2j + y2j )2)′(a20 + b20)xj + aj ≡ 0 (mod pm−n),if aj 6≡ 0, bj 6≡ 0 (mod pm−n);yj ≡ 0, if bj ≡ 0 (mod pm−n);xj ≡ 0, if aj ≡ 0 (mod pm−n),(j = 1, 2, . . . , N).(24)Let t is the number of αj , j = 1, . . . , N , which are coprime with p. Ift = N , then from (23)-(24) we see that every xj deﬁnes the single yjmodulo pm−n, and, hence, there does not exceed pn−(m−n) = p2n−m thevalue yj , corresponding to xj modulo pn.Moreover the secondary congruence in (24) givesxj ≡ a′1ajx1 (mod pm−n), if (a1, p) = 1,oryj ≡ b′1bjy1 (mod pm−n), if (b1, p) = 1.Consequently, the value x1 (or y1, if a1 ≡ 0(mod p)) deﬁnes the singley1 and the pairs (xj , yj) modulo pn−m, j = 2, . . . N . Moreover, N90 Norm Kloosterman sums over Z[i](x1, y1, . . . , xN , yN ) is the solution of the congruences (23). Next, from(24) one easily derives that x1 is the root of the congruenceAx4N+11 ≡ B(mod pm−n). Consequently, the number of solutions of system(23) does not exceed (4N − 1)pN(2n−m).Hence, by (22) we have|K˜N (α0, . . . , αN ; pm, h)| =∑(x1,...,yN )∗∑k∈Z∗pm|epm(−kh+ k′h1)|,where the (∗) indicates that the summation passes over admissible 2N -tuples (x1, y1, . . . , xN , yN ); and h1 is integer, depended from (x1, . . . , yN ).Hence, using an estimate of the rational Kloosterman sum and the estimateof the number of admissible 2N -tuples, we obtain|K˜N (α0, . . . , αN ; pm, h)| 6 2pm2 (4N − 1)pN(2n−m)p2N(m−n) == 2(4N − 1)p2N+12m.(25)In general case, from (22)-(23)|K˜N (α0, α1, . . . , αN ; pm, h)| 6 2pm2+2N(m−n)I(α1, . . . , αN ; pm), (26)where I(α1, . . . , αN ; pm) is the number of solutions of congruence (23).It is easy to see thatI(α1, . . . , αN ; pm) 6 (4N − 1)pmα1+...+mαN .ConclusionIn conclusion note, that the estimate (25) is "the rooted estimate" andapparently in general case is best possibleReferences[1] E. Bombieri, On exponential sums in finite fields,II, Invent. math., 47(1),1978, pp.29-39.[2] R. Bruggeman, Y. Motohashi, Sum formula for Kloosterman sums and fourthmoment of the Dedekind zeta-function over the Gaussian number field, Functioneset Approximatio., XXXI(2003), pp. 23-92.[3] P. Deligne, La conjecture de Weill, I, II, Publ. Math. IHES, 43(1974),pp. 273-307;52(1980), pp. 137-252.[4] B. Dwork, In the rationality of the zeta-function of an algebraic variety, Amer. J.Math. ,82(1960), pp. 631-648.[5] D.R. Heath-Brown, The divisor function d3(n) in arithmetic progressions, ActaArith.,XLVII(1986), pp. 59-86.[6] C. Hooley, An asymptotic formula in the theory of numbers, Proc. London Math.Soc. (3), 7(1957), pp. 396-413.O. Savastru, S. Varbanets 91[7] I.-M. Deshouillers, H. Iwaniec, Kloosterman sums and Fourier coefficients of cuspforms, Invent. math.,70(1982), pp. 219-288.[8] S. P. Varbanets, General Kloosterman sum over ring of Gaussian integers,Ukr.Math. J., 59 (№ 9), 2007, pp. 1179-1201.[9] O. Savastru, P. Varbanets, An additive divisor problem in Z[i], Algebra and discretemathematics, N.1, 2003, pp. 103-110.[10] S. Varbanets, O. Savastru, n-dimensional Kloosterman sums over the ring of thegaussian integers, Vestnik I.I. Mechnikov Odessa National University (to appear).[11] U.B. Zanbyrbaeva, Asympotic problems of number theory in sector region, Disser-tation, Odessa, 1993. (in Russian)[12] N.V. Kuznetsov, Petterson hypothesis for the form with weight zero and Linnikhypothesis, Math. Sb., 3 (1980),pp. 334-383. (in Russian)Contact informationO. Savastru ul.Dvoryanskaya 2, Dept. of computer algebraand discrete mathematics, Odessa national uni-versity, Odessa 65026, UkraineE-Mail: sav−olga@bk.ruS. Varbanets ul.Dvoryanskaya 2, Dept. of computer algebraand discrete mathematics, Odessa national uni-versity, Odessa 65026, UkraineE-Mail: varb@sana.od.uaReceived by the editors: 14.03.2011and in ﬁnal form 05.05.2011.

Norm Kloosterman sums over Z[i]

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Norm Kloosterman sums over Z[i]

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