Gauss and Jacobi Sums, Weil Conjectures

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1 Gauss and Jacobi Sums, Wei Conjectures March 27, 2004 In this note, we define the notions of Gauss and Jacobi sums and ay them to investigate the number of soutions of oynomia euations over finite fieds. Then using them we wi verify the vaidity of the Wei conjectures for a cass of rojective hyersurfaces defined over finite fieds. 1 Trace and Norm in Finite Fieds Throughout this note, excet in the course of the roof of uadratic recirocity aw in Section 4, we assume that is a ower of rime number, and that F k = F k is the uniue finite fied with k eements containing F = F in a fixed agebraic cosure of F. Definition 1.1 For α F k, the trace and norm of α resect to the fied extension F k /F are defined by Tr Fk /F (α := α + α + + α k 1, N Fk /F (α := αα α k 1 resectivey. The foowing emma describes the basic roerties of trace and norm. Lemma 1.2 For α, β F k, and for a F, (a Tr Fk /F (aα + β = atr Fk /F (α + Tr Fk /F (β. (b N Fk /F (αβ = N Fk /F (αn Fk /F (β. (c Tr Fk /F (a = ka and N Fk /F (a = a k. (d Tr Fk /F and Tr Fk /F ma F k onto F. Proof We ony rove the ast one. The fact that α F iff α = α together with the very definition imy that Tr Fk /F (α, N Fk /F (α F. The oynomia x+x + +x k 1 has ess roots in F k than the oynomia x k x, so there exists an α 0 F k such that Tr Fk /F (α 0 = a 0 0. Now for b F given, Tr Fk /F (ba 1 0 α 0 = b. Thus Tr Fk /F : F k F is onto. 1

2 Using the oynomia xx x k 1 and aying a simiar argument wi estabish the surjectivity of N Fk /F : F k F. 2 Gauss Sums This section aims to introduce the imortant notion of a Gauss sum and to estabish its basic roerties. Before we do so, et us reca that: The grou F of (mutiicative characters of F is a cycic grou of order 1 isomorhic to F. For any χ F (abuse of notation!, we extend the domain of definition of χ to whoe F by setting { 1 if χ = ɛ χ(0 := 0 otherwise, where ɛ stands for the trivia character, i.e., ɛ(a = 1 for a a F. Note that by this convention, 1 { 1 if χ = ɛ χ(a = 0 otherwise. a F Definition 2.1 The additive character ψ : F C (see art (a of the foowing roosition is defined by ψ(α := ζ tr(α, where tr = Tr F/F and ζ = e 2πi. Lemma 2.2 (a For α, β F, ψ(α + β = ψ(αψ(β. 1 (b ψ(α 0 1, for some α 0 F. (c α F ψ(α = 0. (d 1 ψ(α(x y = α F { 1 if x = y 0 otherwise. Proof (a Immediate form the definition. (b See the roof of Lemma 1.2. (c Since ψ(α 0 α F ψ(α = α F ψ(α + α 0 = β F ψ(β, by (b we are done. (d Immediate from (c. Definition 2.3 For χ F and α F, the Gauss sum associated to χ (and α is defined by g α (χ := t F χ(tψ(αt. For brevity, we wi denote g 1 (χ by g(χ. 1 It can be shown that for any function ψ : F C satisfying ψ(α+β = ψ(αψ(β, α, β F, there exists an γ F such that ψ(x = ζ tr(γx for a x F. 2

3 In the foowing roosition we wi rove the basic roerties of Gauss sums. 0 if α = 0 and χ ɛ, 0 if α 0 and χ = ɛ, Proosition 2.4 (a g α (χ = if α = 0 and χ = ɛ, χ(α 1 g(χ if α 0 and χ ɛ. (b g(χ = g(χ 1 = χ( 1g(χ. (c If χ ɛ, then g(χg(χ = χ( 1, or euivaenty, g(χ =. Remark For any function f : F C, the Fourier coefficient of f at α F is defined by ˆf(α := 1 f(tψ(αt, and one has the (finite Fourier series exansion of f, namey, t F f(t = α ˆf(αψ(αt. In this terminoogy, the Gauss sum g α (χ is merey the Fourier coefficient of χ at α u to the constant, i.e., g α (χ = χ( α. So, what we are doing here can be transated cometey into the anguage of Fourier anaysis over finite abeian grous (see [N, Chater 4], for a comrehensive account. Proof of the roosition (a Assume that α 0 and χ ɛ. Then we have g α (χ = t F χ(tψ(αt = χ(α 1 t F χ(αtψ(αt = χ(α 1 g(χ. The other arts are obvious. (b Easy! (c Let S = g α (χg α (χ. On the one hand α F S = α 0 χ(α 1 g(χχ(α 1 g(χ = ( 1 g(χ 2. On the other hand S = α = x ( ( χ(xψ(αx χ(yψ( αy y x y ( χ(xχ(y ψ(α(x y = ( 1. α This cometes the roof. 3

4 Gauss sums ay crucia roes in different arts of number theory. For exame, they aear in the functiona euation satisfied by the Dirichet L-functions. f Let χ be a Dirichet character with conductor f, et g(χ := be the (cassica Gauss sum associated to χ, and et L(s, χ := Dirichet L-function attached to χ. Then we have where ( s f π 2 Γ ( s + δ 2 L(s, χ = g(χ ( 1 s f 2 Γ f i δ π δ = Furthermore, one can show that L(1, χ = g(χ f ( 1 s + δ { 0 if χ is even, i.e., χ( 1 = 1 1 if χ is odd, i.e., χ( 1 = 1. L(1, χ = πi g(χ 1 f f f a=1 f χ(aa, if χ( 1 = 1; a=1 2 a=1 n=1 χ(ae 2πia f χ(n n s L(1 s, χ, χ(a og 1 e 2πia f, if χ( 1 = 1, χ χ triv.. For more information, see [K, Chater 2] or [W, Chater 4]. be the Another imortant exame is the Stickeberger theorem about the factorization of Gauss sums in the ring of cycotomic integers Z[ζ m ], ζ m = e 2πi m. Let us exain it recisey. Suose P is an unramified rime in Z[ζ m ], i.e., m P or euivaenty m where Z = P Z. Let F = Z[ζ m] be the (finite residue P fied, and write #F = (= f. It is fairy easy to see that 1 (mod m, that the cosets of 1, ζ m,, ζm m 1 (as eements of F are distinct, and that for any α Z[ζ m ] off P, there is an integer i, uniue mod m, such ( that α 1 m ζ i α m (mod P. We define the m-th ower residue symbo P m as foows ( { α 0 if α P := P ζ i m m if α P. This gives rise to the foowing we-defined mutiicative character for F, χ P (t := ( 1 ( γ γ =, P m P m where γ Z[ζ m ] is an arbitrary reresentative for t F. Corresonding to this character we have the Gauss sum g(χ P. The Stickeberger theorem asserts that 4

5 the rincia idea generated by g(χ P m factors in Z[ζ m ] as (g(χ P m = σt 1 (P t, σ t G where G = Ga (Q(ζ m /Q = { σ t : 1 t m, g.c.d(t, m = 1, σ t (ζ m = ζ t m}. To see a roof of this dee reation and its substantia roe in the roof of the Eisenstein recirocity aw, consut [IR, Chater 14] or [W, Chater 6]. 3 Jacobi Sums Our first objective here is to investigate the number of soutions of (oynomia euations over finite fieds. We wi see that aong the way the notion of a Jacobi sum comes u naturay. To begin with, et s start with the sime euation x m = α. Since the number of soutions of this euation in any finite cycic grou G is the same as the number of soutions for the euation x d = α, where d = g.c.d(m, G, so without oss of generaity and from now on, we assume that m 1. Aso we reca that this number is m, if α is an m-th ower in G; and is 0 otherwise. Lemma 3.1 N(x m = α, the number of soutions of the euation x m = α in F is eua to χ(α. χ m =ɛ Proof For α = 0 the assertion is trivia. So, assume that α F. If α = β m, then χ(α = χ m (β = m = N(x m = α. χ m =ɛ χ m =ɛ Now suose that α is not m-th ower. There is a character χ 1 of order m such that χ 1 (α 1 (for exame the one that takes a given generator of F to e 2πi m works. We have χ(α = χ 1 (α χ(α, and therefore χ m =ɛ χ m =ɛ χ(α = 0 = N(x m = α. χ m =ɛ Next, we wish to evauate N(x m + y m = 1, the number of soutions of the euation x m + y m = 1 in F. By the above emma, we have N(x m + y m = 1 = a+b=1 = a+b=1 N(x m = an(y m = b χ 1 (a χ 2 (b χ m 1 =ɛ χ m 2 =ɛ 5

6 = χ m 1 =χm 2 =ɛ ( a+b=1 χ 1 (aχ 2 (b The above cacuation romts the foowing definition. Definition 3.2 The Jacobi sum attached to χ 1, χ 2 F is defined by J(χ 1, χ 2 := χ 1 (aχ 2 (b. a+b=1 More generay, for χ 1,, χ F, we set J(χ 1,, χ := a 1+ +a =1 χ 1 (a 1 χ (a. It is aso usefu to introduce the foowing sum J 0 (χ 1,, χ := χ 1 (a 1 χ (a. a 1+ +a =0 The foowing summarizes a we need to know about Jacobi sums for the urose of this note. Proosition 3.3 (a If χ 1 = = χ = ɛ, then J(χ 1,, χ = J 0 (χ 1,, χ = 1. (b If χ 1,, χ k ɛ, χ k+1 = = χ = ɛ, then (c If χ ɛ, then J(χ 1,, χ = J 0 (χ 1,, χ = 0. J 0 (χ 1,, χ = χ 1 χ 1 ( 1J(χ 1,, χ 1 s 0 χ 1 χ (s = { χ ( 1( 1J(χ 1,, χ 1 if χ 1 χ = ɛ 0 otherwise. (d If χ 1,, χ, χ 1 χ ɛ, then and therefore J(χ 1,, χ = g(χ 1 g(χ g(χ 1 χ, J(χ 1,, χ = 1 2. (e If χ 1,, χ ɛ, χ 1 χ = ɛ, then J(χ 1,, χ = g(χ 1 g(χ and therefore J(χ 1,, χ = 2 2. = χ ( 1J(χ 1,, χ 1, 6

7 Proof (a Just count the number of summands. (b We have J 0 (χ 1,, χ = χ 1 (a 1 χ k (a k a 1,,a 1 ( ( = k 1 χ 1 (a 1 χ k (a k a 1 a k = 0. And simiar for J(χ 1,, χ. (c For the first euaity, we have J 0 (χ 1,, χ = χ 1 (a 1 χ 1 (a 1 χ (s s 0 a 1+ +a 1 = s = χ 1 χ 1 ( sχ 1 (a 1 χ 1 (a 1 χ (s s 0 a 1 + +a 1 =1 = (χ 1 χ 1 ( sj(χ 1,, χ 1 χ (s s 0 = χ 1 χ 1 ( 1J(χ 1,, χ 1 s 0 χ 1 χ (s. And the second euaity is immediate form the first one. (d First notice that ( ( g(χ 1 g(χ = χ(a 1 ψ(a 1 χ(a ψ(a = a 1 a ( χ 1 (a 1 χ (a ψ(s s a 1+ +a =s = J 0 (χ 1,, χ + J(χ 1,, χ s 0 χ 1 χ (sψ(s = J 0 (χ 1,, χ + J(χ 1,, χ (g(χ 1 χ χ 1 χ (0. ( Now (d foows from ( and (c. (e By what we just roved in (d, g(χ 1 g(χ 1 = g(χ 1 χ 1 J(χ 1,, χ 1. Mutiying both sides of this by g(χ, using χ 1 χ 1 = χ 1 together with the ast art of Proosition 2.4 wi estabish the second euaity of (e. Putting now together this with ( and (c wi resut in the first euaity of (e. 7

8 4 Some Aications This section is devoted to some aications of Gauss and Jacobi sums. Historicay, Gauss sums aeared in Gauss fourth roof of the uadratic recirocity aw in 1811, athough he had worked with them since 1801 and had found some of their basic roerties. Afterwards, Gauss sums were utiized extensivey by Jacobi, Eisenstein, kronecker and others in various roofs of uadratic recirocity aw as we as recirocity aws of higher degrees. Here and for the first aication, we exose an eegant roof of uadratic recirocity aw by means of Gauss and Jacobi sums. Let and be two distinct odd rime numbers, ( and et χ be the uniue character of order 2 on F, i.e., the Legendre symbo. We have J(χ,, χ = }{{} times t 1+ +t =1 χ(t 1 χ(t. If a the t i s are eua, then the corresonding term of the sum has vaue χ ( 1 ( =. And if not, then there are different -tues obtained from ( (t 1,, t by cycic ermutation. This imies that J(χ,, χ (mod, (the congruence reation to be understood in the ring of agebraic integers, and therefore ( J(χ,, χ (mod = 1 1 ( 1 2 g(χ +1 = 1 1 ( ( 1 2 g(χ = 1 1 ( ( a fortiori ( ( = ( = ( ( ( (mod, In the seue of this section we derive a formua, and through that, an estimate for N = N(a 1 x m a x m = b, the number of soutions of the euation a 1 x m a x m = b, a i F. Moreover, we wi find, as a by-roduct, the number of rojective oints on the hyersurface defined by a 0 x m 0 + a x m = 0 in P F. 8

9 As before there wi be no restriction if we assume that m i 1. Theorem 4.1 (a If b = 0, then N = 1 + χ 1 (a 1 1 χ (a 1 J 0 (χ 1,, χ. (1 The sum is over a -tues of characters χ 1,, χ, where χ i ɛ, χ mi i = ɛ, and χ 1 χ = ɛ. If M stands for the number of such -tues, then N 1 M( (2 (b If b 0, then N = 1 + χ 1 χ (bχ 1 (a 1 1 χ (a 1 J(χ 1,, χ, (3 where χ i ɛ, χ mi i ɛ, then = ɛ. If M 1 denotes the number of such -tues with χ 1 χ N 1 M M (4 Proof Note that N = = a 1u 1+ +a u =b χ 1,,χ ord(χ i m i ( N(x m1 1 = u 1 N(x m = u a 1u 1+ +a u =b χ 1 (u 1 χ (u. If b = 0, the inner sum is and if b 0, it is χ 1 (a 1 1 χ (a 1 J 0 (χ 1,, χ ; χ 1 χ (bχ 1 (a 1 1 χ (a 1 J(χ 1,, χ. Now (1 and (3 wi foow from arts (a, (b and (c of the Proosition 3.3. Invoking arts (d and (e of the same roosition wi estabish the roof of (2 and (4. Remark In the roof of Wei conjectures, we wi see that M = 1 m ( (m ( 1 +1 (m 1 and M 1 = (m 1 +1 M. Coroary 4.2 The number of the oints (in P F on the hyersurface defined by a 0 x m 0 + a x m = 0 (a i F is eua to where χ i ɛ, χ m i = ɛ and χ 0 χ = ɛ. χ 0,,χ χ 0 (a 1 0 χ (a 1 g(χ 0 g(χ 9

10 Proof By revious theorem, the desired number is eua to 1 ( + χ 0 (a χ (a 1 J 0 (χ 0,, χ 1. However, by arts (c and (d of Proosition 3.3 and by art (c of Proosition 2.4, we have 1 1 J 0(χ 0,, χ = χ 0 ( 1J(χ 1,, χ = χ 0 ( 1 g(χ 0g(χ 1 g(χ g(χ 0 g(χ 1 χ = g(χ 0 g(χ. This cometes the roof. 5 Wei Conjectures In this section we wi recover the Wei conjectures for the hyersurface H defined by a 0 x m 0 + a x m = 0. So, et us first reca the definition of Z(H/F, T. For any k 1, et N k denote the number of oints on H defined over F k. The zeta function Z(H/F, T of H is defined by ( T k Z(H/F, T := ex N k, k where ex(u := 1 + u + u2 2! + u3 3! +. Lemma 5.1 We have k=1 Z(H/F, T = (1 α 1T (1 α r T (1 β 1 T (1 β s T, α i, β j C iff for k = 1, 2, 3,. N k = β k β k s α k 1 α k r Proof On the one hand, og Z(H/F, T = k=1 N k T k k. 10

11 And on the other hand, og (1 α 1T (1 α r T (1 β 1 T (1 β s T = og(1 β j T + og(1 α i T j i = βj k α k T k i k. j i k=1 This cometes the roof. Before we roceed to state and rove of Wei conjectures, we need to investigate the reation between the characters of F with those of F k. The key ink wi be rovided by the norm function N Fk /F. Let χ F, and set χ (k = χ o N FK /F, i.e., χ (k (a = χ(n Fk /F (a for a F k. The foowing affirmations can be easiy verified: 1. χ (k F k. 2. χ (k 1 = χ (k 2 iff χ 1 = χ ord(χ (k m iff ord(χ m. 4. χ (k (a = χ(a k for a a F. It immediatey foows from 3 that if χ runs the set of characters of order dividing m (in F, then χ (k wi do the same in F k. And finay, the foowing cassica resut the interreation between Gauss sums g(χ and g(χ (k wi rovide our ast ingredient. Theorem 5.2 (Hasse-Davenort With the above notations, g(χ (k = ( 1 k+1 g(χ k. Proof See [IR, Chater 11], for an eegant roof. Chater 2], as a ong exercise. It is aso outined in [K, Now we have a necessary toos to rove the foowing secia case of the Wei conjectures. Theorem 5.3 (Wei The zeta function Z(H/F, T of the hyersurface H defined by the euation a 0 x m 0 + a x m = 0 has the foowing roerties: (a (Rationaity Z(H/F, T = where P (T is the oynomia P (T ( 1 (1 T (1 T (1 1 T, ( 1 ( χ 0(a 1 0 χ (a 1 g(χ 0 g(χ T. 11

12 The characters χ i are subject to the conditions χ i ɛ, χ m i = ɛ, and χ 0 χ = ɛ. Moreover, we caim that P (T Z[T ]. (b The degree of P (T is eua to d = 1 ( (m ( 1 +1 (m 1. m (c (Functiona Euation The maing α 1 is a bijection of the 1 α set of zeros of P (T. Euivaenty, Z(H/F, T satisfies the foowing functiona euation where Z(H/F, 1 ( 1 1 = ωt d 1 2 ( 2( 1d Z(H/F, T, T ω = ( 1 d P (α=0 α ( 1. (d The recirocas of zeros of P (T are agebraic integers. (e (Riemann Hyothesis The zeros of P (T have absoute vaue 1 2. Proof (a By Coroary 4.2, N k is eua to k( k k χ (k 0,,χ(k χ (k 0 (a 1 0 χ(k (a 1 g(χ (k 0 g(χ(k, where χ (k i ɛ, χ (km i = ɛ, χ (k 0 χ (k = ɛ (we are using ɛ simutaneousy for the trivia character of a F k s. So, by 3 and 4 of the above and by Hasse- Davenort reation, we infer that N k = = 1 ik + 1 k i=0 χ 0,,χ 1 ( i k + ( 1 +1 i=0 ( ( 1 (k+1(+1 χ 0 (a 1 0 k χ (a 1 k g(χ 0 k g(χ k χ 0,,χ ( ( 1 +1 χ 0 (a 1 0 χ (a 1 g(χ 0 g(χ k. This roves (a, excet the fact that P (T is in Z[T ]. We ostone its roof unti art (d. (b Set A = {(χ 0,, χ : χ i ɛ, χ m i = ɛ, χ 0 χ = ɛ}, B = {(χ 0,, χ : χ i ɛ, χ m i = ɛ, χ 0 χ ɛ}. Ceary deg P (T = A. Since the ma (χ 0,, χ (χ 0,, χ 1 is a bijection between A and B 1, we get A + A +1 = A + B = (m

13 Using this and the initia A 1 = m 1, (b wi foow by a sime induction. (c To rove the first statement it is sufficient to show that α 1 1 α is a we-defined ma from the set of zeros of P (T to itsef. So, et α = ( 1 +1 χ 0 (a 0 χ (a g(χ 0 1 g(χ 1 be a zero of P (T corresonding to (χ 0,, χ A. We have 1 α = ( 1 +1 χ 0 (a 0 χ (a g(χ 0 1 g(χ 1 = ( χ 1 0 (a 1 0 χ 1 = ( χ 1 0 (a 1 = ( χ 1 0 (a 1 0 χ 1 0 χ 1 (a 1 g(χ 0 g(χ (a 1 χ 0 ( 1g(χ 1 0 χ ( 1g(χ 1 (a 1 g(χ 1 0 g(χ 1. and the ast uantity is the reciroca of that zero of P (T which is corresonding to (χ 1 0,, χ 1 A. Checking that Z(H/F, T satisfies the aforesaid functiona euation is a very straightforward but somehow tedious cacuation, eft to the reader. (d Obviousy, the vaues of any character are agebraic integers. In fact, if χ m = ɛ, then every vaue that χ takes is a unit in Z[ζ m ]. Therefore, by art (e of Proosition 3.3, if (χ 0,, χ A, then ( χ 0(a 1 0 χ (a 1 g(χ 0 g(χ = ( 1 +1 χ 1 0 (a 0 χ 1 (a χ ( 1J(χ 0,, χ 1 Z[ζ m ]. Now we accomish the roof of art (a by showing that P (T Z[T ]. From the rationaity of the zeta function and working inside the fied Q[[T ]], it is amost cear that P (T Q[T ]. (notice it is immediate from the very definition that Z(H/F, T Q[[T ]]. On the other hand, (d imies that P (T A[T ], where A is the ring of agebraic integers. Hence, the coefficients of P (T are in Q A = Z. (e Immediatey from art (c of Proosition 2.4 we deduce that 1 ( 1+1 χ 0(a 1 0 χ (a 1 g(χ 0 g(χ = +1 2 = 1 2, which is euivaent to the desired statement. 13

14 References [IR] K. Ireand and M. Rosen, A Cassica Introduction to Modern Number Theory, Graduate Texts in Mathematics 52, Sringer-Verag, New York, [K] N. Kobits, Introduction to Eitic Curves and Moduar Forms, Graduate Texts in Mathematics 97, Sringer-Verag, [N] Mevyn B. Nathanson, Eementary Methods in Number Theory, Graduate Texts in Mathematics 195, Sringer-Verag, [W] Lawrence C. Washington, Introduction to Cycotomic Fieds, Graduate Texts in Mathematics 83, Sringer-Verag,