BINOMIAL CHARACTER SUMS MODULO PRIME POWERS

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1 BINOMIAL CHARACTER SUMS MODULO PRIME POWERS VINCENT PIGNO AND CHRISTOPHER PINNER Résumé. On montre que les sommes binomiales et liées de caractères multilicatifs χx l Ax k + B w, x,=1 ont une évaluation simle our m suffisamment grand our m si ABk. χ 1 xχ Ax k + B, Abstract. We show that the binomial and related multilicative character sums χx l Ax k + B w, x,=1 have a simle evaluation for large enough m for m if ABk. χ 1 xχ Ax k + B, 1. Introduction For a multilicative character χ mod m, and rational functions fx, gx Zx one can define the mixed exonential sum, 1 Sχ, gx, fx, m := m χgxe mfx where e y x = e πix/y and indicates that we omit any x roducing a non-invertible denominator in f or g. When m = 1 such sums have Weil [15] tye bounds; for examle if f is a olynomial and the sum is non-degenerate then Sχ, gx, fx, degf + l 1 1/, where l denotes the number of zeros and oles of g see Castro & Moreno [] or Cochrane & Pinner [5] for a treatment of the general case. When m methods of Cochrane and Zheng [3] see also [6] & [7] can be used to reduce and simlify the sums. For examle we showed in [1] that the sums 3 χxe mnx k Date: 1th Aril Mathematics Subject Classification. Primary:11L10, 11L40; Secondary:11L03,11L05. Key words and hrases. Character Sums, Gauss sums, Jacobi Sums. 1

2 VINCENT PIGNO AND CHRISTOPHER PINNER can be evaluated exlicitly when m is sufficently large for m if nk. We show here that the multilicative character sums, 4 S χ, x l Ax k + B w, m = x χx l Ax k + B w similarly have a simle evaluation for large enough m for m if ABk. Equivalently, for characters χ 1 and χ mod m we define 5 Sχ 1, χ, Ax k + B, m = χ 1 xχ Ax k + B. These include the mod m generalizations of the classical Jacobi sums 6 Jχ 1, χ, m = χ 1 xχ 1 x. These sums have been evaluated exactly by Zhang Weneng & Weili Yao [18] when χ 1, χ and χ 1 χ are rimitive and m is even some generalizations are considered in [0]. Writing 7 χ 1 = χ l, χ = χ w, χ 1 xχ Ax k + B = χx l Ax k + B w, with χ 1 = χ 0 the rincial character if l = 0, the corresondence between 4 and 5 is clear. Of course the restriction x in 4 only differs from when l = 0. We shall assume throughout that χ is a rimitive character mod m equivalently χ is rimitive and w; if χ is not rimitive but χ 1 is rimitive then Sχ 1, χ, Ax k + B, m = 0 since χ 1x+y m 1 = 0, if both are not rimitive we can reduce to a lower modulus Sχ 1, χ, Ax k + B, m = Sχ 1, χ, Ax k + B, m 1. It is interesting that the sums 3 and 4 can both be written exlicitly in terms of classical Gauss sums for any m 1. In articular one can trivially recover the Weil bound in these cases. We exlore this in Section. We assume, noting the corresondence 7 between 4 and 5, that 8 gx = x l Ax k + B w, w where k, l are integers with k > 0 else x x 1 and A, B non-zero integers with 9 A = n A, 0 n < m, A B. We define the integers d 1 and t 0 by 10 d = k, 1, t k. For m n + t + 1 it transires that the sum in 4 or 5 is zero unless 11 χ 1 = χ k 3, for some mod m character, χ 3 i.e. χ is the k, φ m /k, l, φ m th ower of a character, and we have a solution, x 0, to a characteristic equation of the form, 1 g m+n min{m 1, [ x 0 mod ]+t} with 13 x 0 Ax k 0 + B.

3 BINOMIAL CHARACTER SUMS MODULO PRIME POWERS 3 Notice that in order to have a solution to 1 we must have 14 n+t l, t l + wk, if m > t + n + 1 equivalently χ 1 is induced by a rimitive mod m n t character and χ 1 χ w is a rimitive mod m t character and n+t l if m = t + n + 1. When 11 holds, 1 has a solution x 0 satisfying 13 and m > n + t + 1, Theorem 3.1 below gives an exlicit evaluation of the sum 5. From this we see that 15 { χ 1 xχ Ax k + B = d m 1, if t + n + 1 < m t + n +, d m+n +t, if t + n + < m. The condition m > t + n + 1 is natural here; if t m n then one can of course use Euler s Theorem to reduce the ower of in k to t = m n 1. If t = m n 1 and the sum is non-zero then, as in a Heilbronn sum, we obtain a mod sum, m 1 1 χxl Ax k + B w, where one does not exect a nice evaluation. For t = 0 the result 15 can be obtained from [3] by showing equality in their S α evaluated at the d critical oints α. For t > 0 the α will not have multilicity one as needed in [3]. Condition 11 will arise naturally in our roof of Theorem 3.1 but can also be seen from elementary considerations. Lemma 1.1. For any odd rime, multilicative characters χ 1, χ mod m, and f 1, f in Z[x], the sum S = for some mod m character χ 3. χ 1 xχ f 1 x k e mf x k is zero unless χ 1 = χ k 3 Proof. Taking z = a φm /k,φ m, a a rimitive root mod m, we have z k = 1 and S = χ 1 xzχ f 1 xz k e mf xz k = χ 1 zs. Hence if S 0 we must have 1 = χ 1 z = χ 1 a φm /k,φ m and χ 1 a = e φ m c k, φ m for some integer c. For an integer c 1 satisfying c k, φ m c 1 k mod φ m, we equivalently have χ 1 = χ k 3 where χ 3 a = e φm c 1. Finally we observe that if χ is a mod rs character with r, s = 1, then χ = χ 1 χ for a mod r character χ 1 and mod s character χ, and for any gx in Z[x] rs r s χgx = χ 1 gx χ gx. Thus it is enough to work modulo rime owers.. Gauss Sums and Weil tye bounds For a character χ mod j, j 1, we let Gχ, j denote the classical Gauss sum Gχ, j = χxe j x. j

4 4 VINCENT PIGNO AND CHRISTOPHER PINNER Recall see for examle Section 1.6 of Berndt, Evans & Williams [1] that 16 Gχ, j j/, if χ is rimitive mod j, = 1, if χ = χ 0 and j = 1, 0, otherwise. It is well known that the mod Jacobi sums 6 and their generalization to finite fields can be written in terms of Gauss sums see for examle Theorem.1.3 of [1] or Theorem 5.1 of [10]. This extends to the mod m sums. For examle when χ 1, χ and χ 1 χ are rimitive mod m 17 Jχ 1, χ, m = Gχ 1, m Gχ, m Gχ 1 χ, m, and Jχ 1, χ, m = m/ see Lemma 1 of [19] or [0]; the relationshi for Jacobi sums over more general residue rings modulo rime owers can be found in [14]. We showed in [1] that for n the sums Sχ, x, nx k, m = χxe mnx k are zero unless χ = χ k 1 for some character χ 1 mod m, in which case summing over the characters whose order divides k, φ m to ick out the kth owers Sχ, x, nx k, m = χ 1 χ ngχ 1 χ, m. χ k,φm =χ 0 From this one immediately obtains a Weil tye bound Sχ, x, nx k, m k, φ m m/. From Lemma 1.1 we know that the sum in 5 is zero unless χ 1 = χ k 3 for some character χ 3 mod m, in which case the sum can be written as k, φ m mod m Jacobi like sums m χ 5xχ Ax + B and again be exressed in terms of Gauss sums. Theorem.1. Let be an odd rime. If χ 1, χ are characters mod m with χ rimitive and χ 1 = χ k 3 for some character χ 3 mod m, and n and A are as defined in 9, then χ 1 xχ Ax k +B = n χ 4 X where X denotes the mod m characters χ 4 with χ D 4 that χ 3 χ 4 is a mod m n character. χ 3 χ 4 A χ χ 3 χ 4 B Gχ 3χ 4, m n Gχ χ 3 χ 4, m Gχ, m, We immediately obtain the Weil tye bound 18 Sχ1, χ, Ax k + B, m k, φ m m+n/. For m = 1 and A this gives us the bound 1 χ x l Ax k + B w d 1, = χ 0, D = k, φ m, such

5 BINOMIAL CHARACTER SUMS MODULO PRIME POWERS 5 where d = k, 1. For l = 0 we can slightly imrove this for the comlete sum, 1 χax k + B d 1 1, x=0 since, taking χ 1 = χ 3 = χ 0, χ = χ, the χ 4 = χ 0 term in Theorem.1 equals χb, the missing x = 0 term in 4. These corresond to the classical Weil bound after an aroriate change of variables to relace k by d. For m t + 1 the bound 18 is d m+n +t, so by 15 we have equality in 18 for m n + t +, but not for t + n + 1 < m < t + n +. Notice that if k, φ m = 1, as in the generalized Jacobi sums 6, with χ rimitive, and χ 1 = χ k 3 is a mod m n character if A, then we have the single χ 4 = χ 0 term and χ 1 xχ Ax k + B = n χ 3 A χ χ 3 B Gχ 3, m n Gχ χ 3, m Gχ, m, of absolute value m+n/ if χ, χ χ 3 and χ 3 are rimitive mod m and m n noting that Gχ, m = χ 1Gχ, m we lainly recover the form 17 in that case. For the multilicative analogue of the classical Kloostermann sums, χ assumed rimitive and A, Theorem.1 gives a sum of two terms of size m/ m χax + x 1 = χ 3A Gχ Gχ, m 3, m + χ AG χ 3 χ, m when χ = χ 3 otherwise the sum is zero, where χ denotes the mod m extension of the Legendre symbol taking χ = χ, χ 1 = χ, k = we have D = and χ 4 = χ 0 or χ. For m = 1 this is Han Di s [8, Lemma 1]. Cases where we can write the exonential sum exlicitly in terms of Gauss sums seem rare. Best known after the quadratic Gauss sums are erhas the Salié sums, evaluated by Salié [13] for m = 1 see Williams [16],[17] or Mordell [11] for a short roof and Cochrane & Zheng [4, 5] for m ; for AB { m χ xe max+bx 1 = χ 1 m 1 e mγ + e m γg χ,, m odd, B 1 m χ γe mγ + χ γe m γ, m even, if AB = γ mod m, and zero if χ AB = 1. Cochrane & Zheng s m method works with a general χ as long as their critical oint quadratic congruence does not have a reeat root, but formulae seem lacking when m = 1 and χ χ. For the Jacobsthal sums we get essentially Theorems & of [1] 1 m=1 m m k + B = 1 m=0 m k + B = B k 1 j=0 k 1 B j=1 χb j+1 Gχj+1, Gχ j+1 χ, Gχ,, χb j Gχj, Gχ j χ, Gχ,, when 1 mod k and B, where χ denotes a mod character of order k and χ the mod character corresonding to the Legendre symbol see also [9].

6 6 VINCENT PIGNO AND CHRISTOPHER PINNER Proof. Observe that if χ is a rimitive character mod j, j 1, then 19 χye j Ay = χagχ, j. j Indeed, for A this is lain from y A 1 y. If A and j = 1 the sum equals χy = 0 and for j writing y = au+φj 1 v, a a rimitive root mod m, χa = e φ j c, u = 1,..., φ j 1, v = 1,..,, 0 χye j Ay = j φ j 1 χa u e j Aa u Hence if χ is a rimitive character mod m we have Gχ, m χ Ax k + B = and, since χ 1 = χ k 3 and D = k, φ m, Gχ, m Since B we have χ 1 xχ Ax k + B = = = χ 3 x k χ 3 x D χ D 4 =χ0 = χ D 4 =χ0 = χ D 4 =χ0 e cv = 0. v=1 χ ye max k + By χ ye max k + By χ ye max D + By χ 3 uχ 4 u χ ye mby χ ye mau + By χ χ 3 χ 4 ye mby χ χ 3 χ 4 ye mby = χ χ 3 χ 4 BGχ χ 3 χ 4, m. If χ 3 χ 4 is a mod m n character then χ 3 χ 4 ue mauy χ 3 χ 4 ue mau. m n χ 3 χ 4 ue mau = n χ 3 χ 4 ue m na u = n χ 3 χ 4 A Gχ 3 χ 4, m n. If χ 3 χ 4 is a rimitive character mod j for some m n < j m then by 0 and the result follows. j χ 3 χ 4 ue mau = m j χ 3 χ 4 ue j j m n A u = 0,

7 BINOMIAL CHARACTER SUMS MODULO PRIME POWERS 7 Notice that if m n+ then by 16 the set X can be further restricted to those χ 4 with χ 3 χ 4 rimitive mod m n. Hence if t k, with m n + t + and we write χ 3 a = e φ m c 3, χ 4 a = e φ m c 4 we have m 1 t c 4, n c 3 + c 4, giving n c 3. From χ k 3 = χ 1 = χ l this yields n+t c 3 k = c 1 = cl and n+t l. If n > 0 we deduce that t l + wk. Moreover when n = 0 reversing the roles of A and B gives t l + wk. Hence when m n + t + we have Sχ 1, χ, Ax k + B, m = 0 unless 14 holds. For m = n + t + 1 we similarly still have n+t l. 3. Evaluation of the Sums Theorem 3.1. Suose that is an odd rime and χ 1, χ are mod m characters with χ rimitive. If χ satisfies 11, and 1 has a solution x 0 satisfying 13, then m m 1, if t + n + 1 < m t + n +, χ 1 xχ Ax k + B = dχgx 0 m+n +t, if m > t + n +, m n even, +t ε 1, if m > t + n +, m n odd, m+n where n, d, t and g are as defined in 9, 10 and 8, with { α ε 1 = e β α mod 4, ε, ε = i 3 mod 4, where α and β are integers defined in 9 below and α is the Legendre symbol. If χ does not satisfy 11, or 1 has no solution satisfying 13, then the sum is zero. For the mod m Jacobi sums, χ 1 = χ l, χ = χ w, χ rimitive mod m with lwl + w, we have x 0 = ll + w 1 and { m χ 1 xχ 1 x = χ 1lχ w 1, if m is even, χ 1 χ l + w m with r and c as in 1 and 3 below. rc Proof. Let a be a rimitive root mod such that 1 a φ = 1 + r, r. lwl+w ε, if m 3 is odd, Thus a is a rimitive root for all owers of. We define the integers r l, r l, by a φl = 1 + r l l, so that r = r 1. Since 1 + r s+1 s+1 = 1 + r s s, for any s 1 we have r s+1 r s mod s. We define the integers c, c 1 = cl, c = cw, by 3 χa = e φ m c, χ 1 a = e φ m c 1, χ a = e φ m c. Since χ is assumed rimitive we have c. We write { γ = u φl 1, if m n + t +, + v, L := d t, if m > n + t +, m n

8 8 VINCENT PIGNO AND CHRISTOPHER PINNER and observe that if u = 1,..., d m L and v runs through an interval I of length φ L /d then γ runs through a comlete set of residues mod φ m. Hence setting hx = Ax k + B and writing x = a γ we have m d m 1 χ 1 a v χ 1 a u φl d χ h a u φl d +v. χ 1 xχ hx = v I Since L + t + n m we can write h a u φl d +v = A a k u φl+t d t a vk + B = A 1 + r L+t L+t k u d t a vk + B k ha v + A u d t a vk r L+t L+t+n mod m. This is zero mod if ha v and consequently any such v give no contribution to the sum. If ha v then, since r L+t r L+t+n mod L+t, h a u φl d +v k ha v 1 + A u d t ha v 1 a vk r L+t+n L+t+n mod m ha v a A k u ha d t v 1 a vk φ L+t+n mod m. Thus, χ 1 xχ hx = v I ha v d m L χ 1 a v χ ha v χ 1 a u φl d χ a u φl d Aka vk ha v 1, where the inner sum d m L e d m L u c1 + c Aha v 1 ka vk is d m L if k 4 c 1 + c ha v 1 A a vk d t d t+n 0 mod d m L and zero otherwise. Thus our sum will be zero unless 4 has a solution with ha v. For m n + t + 1 we have m L t + n and a solution to 4 necessitates d t+n c 1 giving us condition 11 with t+n l for m > n + t + 1. Hence for m > n + t + 1 we can simlify the congruence to 5 ha v c1 k d t+n + c A a vk d t 0 mod m L t n and for a solution we must have t c 1 + kc. Equivalently, 6 cg a v d t+n 0 mod m t n L, and the characteristic equation 1 must have a solution satisfying 13. Suose that 1 has a solution x 0 = a v0 with hx 0 and that m > n + t + 1. Rewriting the congruence 6 in terms of the rimitive root, a, gives a vk a b mod m t n L

9 BINOMIAL CHARACTER SUMS MODULO PRIME POWERS 9 for some integer b. Thus two solutions to 6, a v1 and a v must satisfy v 1 k v k mod φ m t n L. That is v 1 v mod 1 d if m n + t + and if m > n + t + v 1 v mod φm n t L d where m n t L = L if m n is even and L 1 if m n is odd. Thus if n + t + 1 < m n + t + or m > n + t + and m n is even our interval I contains exactly one solution v. Choosing I to contain v 0 we get that χ 1 xχ hx = d m L χ 1 x 0 χ hx 0. Suose that m > n + t + with m n odd and set s := m n 1. In this case I will have solutions and we ick our interval I to contain the solutions v 0 + y s t 1 1 d where y = 0,..., 1. Since d t c 1 and d t k we can write, with g defined as in 8, g 1 x := gx c = x c1 Ax k + B c =: H x dt. Thus, setting χ = χ c 4, where χ 4 is the mod m character with χ 4 a = e φm 1, where χ 1 xχ hx = d m+n 1 = d m+n 1 1 +t y=0 1 +t y=0 χ g a v0+ys t 1 1 d χ 4 H x dt 0 a yφs, 7 x dt 0 a yφs = x dt r s s y = x dt 0 + yr s x dt 0 s mod m n 1. Since x dt 0 n H x dt = xg 1 x d t+n x dt Z[x], we have n Hk k!, for all k 1. As xg 1x = c 1 +kc g 1 x c kbg 1 x/hx, n H c1 x dt x dt = d t + c k d t c k B xg d t hx 1 1 x k d t+n +c d t A Bx k g 1x hx. Plainly a solution x 0 to 1 satisfying 13 also has g 1x 0 0 mod m+n 1 +t and 8 x 0 g 1x 0 d t+n for some integer λ, and = λ m n 1, H x dt 0 = x dt 0 λ m+n 1, n H x dt 0 c k d t A Bx k dt 0 g 1 x 0 mod. hx 0

10 10 VINCENT PIGNO AND CHRISTOPHER PINNER Hence by the Taylor exansion, using 7 and that r s r m 1 r mod, H x dt 0 a yφs Hx dt 0 + H x dt 0 yr s x dt 0 m n H x dt 0 y rsx dt 0 m n 1 mod m g 1 x βy + αy r m 1 m 1 mod m g 1 x 0 a βy+αy φ m 1 mod m, with 9 β := g 1 x 0 1 λ, α := 1 c hx 0 ra B and k d t x k 0, χ 4 H x dt 0 a yφs = χgx 0 e αy + βy. Since lainly α, comleting the square then gives the result claimed χ 1 xχ hx = d m+n 1 +t χgx 0 e 4 1 α 1 β = d m+n 1 +t χgx 0 e 4 1 α 1 β e αy y=0 α ε 1 where ε is 1 or i as is 1 or 3 mod 4. Notice that if x 0 is a solution to the stronger congruence g x 0 0 mod [ m+n ]+t+1 then β = 0 and the e 4 1 α 1 β can be omitted. References [1] B.C. Berndt, R.J. Evans & K.S. Williams, Gauss and Jacobi Sums, Canadian Math. Soc. series of monograhs and advanced texts, vol. 1, Wiley, New York [] F. Castro & C. Moreno, Mixed exonential sums over finite fields, Proc. Amer. Math. Soc , [3] T. Cochrane, Exonential sums modulo rime owers, Acta Arith , no., [4] T. Cochrane, Exonential sums with rational function entries, Acta Arith , no. 1, [5] T. Cochrane and C. Pinner, Using Steanov s method for exonential sums involving rational functions, J. Number Theory , no., [6] T. Cochrane, Zhiyong Zheng, Pure and mixed exonential sums, Acta Arith , no. 3, [7] T. Cochrane, Zhiyong Zheng, A survey on ure and mixed exonential sums modulo rime owers, Number theory for the millennium, I Urbana, IL, 000, , A K Peters, Natick,MA, 00. [8] Han Di, A hybrid mean value involving two-term exonential sums and olynomial character sums, to aear Czech. Math. J. [9] P. Leonard & K. Williams, Evaluation of certain Jacobsthal sums, Boll. Unione Mat. Ital , [10] R. Lidl & H. Niederreiter, Finite Fields, Encycloedia of Mathematics and its alications 0, nd edition, Cambridge University Press, [11] L. J. Mordell, On Salié s sum, Glasgow Math. J , 5-6. [1] V. Pigno & C. Pinner, Twisted monomial Gauss sums modulo rime owers, submitted to Acta Arith. [13] Hans Salié, Uber die Kloostermanschen summen Su, v; q, Math. Zeit , [14] J. Wang, On the Jacobi sums mod P n, J. Number Theory , [15] A. Weil, On some exonential sums, Proc. Nat. Acad. Sci. U.S.A , [16] K. Williams, On Salié s sum, J. Number Theory ,

11 BINOMIAL CHARACTER SUMS MODULO PRIME POWERS 11 [17] K. Williams, Note on Salié s sum, Proc. Amer. Math. Soc., Vol 30, no. 1971, [18] W. Zhang & W. Yao, A note on the Dirichlet characters of olynomials, Acta Arith , no. 3, 5-9. [19] W. Zhang & Y. Yi, On Dirichlet Characters of Polynomials, Bull. London Math. Soc , no. 4, [0] W. Zhang & Z. Xu, On the Dirichlet characters of olynomials in several variables, Acta Arith , no., Deartment of Mathematics, Kansas State University, Manhattan, KS address: ignov@math.ksu.edu Deartment of Mathematics, Kansas State University, and Manhattan, KS address: inner@math.ksu.edu