L-FUNCTIONS FOR GAUSS AND JACOBI SUMS

Transcription

1 L-FUNCTIONS FOR GAUSS AND JACOBI SUMS KEITH CONRAD 1. Itroductio For a multiplicative character χ: F q C ad additive character ψ : F q C o a fiite field F q of order q, their Gauss sum is G(χ, ψ) = χ(c)ψ(c), c F q where we exted χ to 0 by χ(0) = 0. Here are two fudametal properties of Gauss sums. (1) For otrivial χ ad ψ, G(χ, ψ) = q. (This is ot true if oe of the characters is trivial: if χ is trivial ad ψ is ot the G(χ, ψ) = 1, if ψ is trivial ad χ is ot the G(χ, ψ) = 0, ad if χ ad ψ are both trivial the G(χ, ψ) = q 1.) (2) (Hasse Daveport) For 1 let χ = χ N Fq /F q ad ψ = ψ Tr Fq /F q be the liftigs of χ ad ψ to multiplicative ad additive characters o F q. The G(χ, ψ ) = ( G(χ, ψ)). (This suggests G(χ, ψ) is more fudametal.) We will show how both properties of Gauss sums ca be iterpreted as properties of L-fuctios o F q [T ]: the first property says a certai L-fuctio satisfies the Riema hypothesis ad the secod property follows from comparig the additive (Dirichlet series) ad multiplicative (Euler product) represetatios of a L-fuctio. Aalogous results for Jacobi sums, based o the same ideas, are sketched at the ed. 2. Gauss sums ad the Riema Hypothesis Dirichlet characters are group homomorphisms (Z/m) C ad have L-fuctios. For ocostat M i F q [T ], the fiite group (F q [T ]/M) is aalogous to (Z/m) ad we call ay homomorphism η : (F q [T ]/M) C a character mod M. Exted η to 0 by η(0) = 0 ad lift η to F q [T ] by declarig η(a) = η(a mod M). This fuctio η o F q [T ] is totally multiplicative, ad by aalogy to the defiitio of the L-fuctio of a Dirichlet character we defie the L-fuctio of η to be L(s, η) := η(a) N(A) s = η(a) 1 q s 0 moic A deg A= for Re(s) > 1, where the ier sum rus over moic A of degree ad N(A) = F q [T ]/A = q deg A. Note the costat term of L(s, η) is 1 (occurrig for A = 1). By the chage of variables u = 1/q s we ca view L(s, η) as a formal power series i u: L(u, η) := η(a) u, so L(s, η) = L(1/q s, η). moic A η(a)u deg A = 0 deg A= Theorem 2.1. If η is otrivial the for deg M the coefficiet of u vaishes. 1

2 2 KEITH CONRAD Proof. (This proof is take from [2, p. 36].) For each moic A of degree, write A = MQ+R for Q, R F q [T ] with R = 0 or deg R < deg M. Sice A is moic of degree, Q is moic of degree deg M. By uiqueess of the quotiet ad remaider for each A, as A rus over all moics of degree the pair (Q, R) rus over all pairs of a moic Q of degree deg M ad a polyomial R of degree less tha deg M (icludig R = 0). Therefore η(r) deg A= η(a) = Q,R η(mq + R) = Q,R η(r) = q deg M R sice there are q deg M choices of Q. Sice R is ruig over the polyomials of degree less tha M alog with 0, which represets all of F q [T ]/M, ad η vaishes o polyomials havig a factor i commo with M, we have η(r) = η(r) = 0 R R (F q[t ]/M) because the sum of a otrivial character over a fiite abelia group is 0. Now focus o the case deg M = 2. For otrivial η the coefficiet of u is 0 if 2, so (2.1) L(u, η) = 1 + η(t + c) u. c F q We will see that whe M = T 2, the coefficiet of u here is essetially a Gauss sum. Theorem 2.2. The characters of (F q [T ]/T 2 ) are pairs of a multiplicative ad additive character o F q. Proof. We uwid what the elemets of (F q [T ]/T 2 ) look like. To say a + bt mod T 2 is ivertible meas a 0. By rewritig b as ab we ca write the ivertible elemets as a(1 + bt ) mod T 2 for a F q ad b F q. Sice we have a isomorphism a(1 + bt )a (1 + b T ) aa (1 + (b + b )T ) mod T 2, (F q [T ]/T 2 ) = F q F q by a(1 + bt ) mod T 2 (a, b). Therefore the character group of (F q [T ]/T 2 ) is the pairs (χ, ψ) for a multiplicative character χ: F q C ad a additive character ψ : F q C : (2.2) a(1 + bt ) mod T 2 χ(a)ψ(b). (Sayig ψ is trivial is the same as sayig this character mod T 2 ca be defied modulo T, ad thus is ot primitive mod T 2.) Returig to (2.1), the liear polyomials T + c relatively prime to T 2 are those with c 0, i which case T + c = c(1 + (1/c)T ), so if the character η o (F q [T ]/T 2 ) is realized by (2.2) with χ or ψ otrivial, so η is otrivial, the the L-fuctio of η is 1 + χ(c)ψ(1/c) u = 1 + χ(1/c)ψ(c) u = 1 + χ(c)ψ(c) u = 1 + G(χ, ψ)u.

3 L-FUNCTIONS FOR GAUSS AND JACOBI SUMS 3 Replacig χ with χ ad u with 1/q s, the L-fuctio of the character a(1 + bt ) mod T 2 χ(a)ψ(b) o (F q [T ]/T 2 ) for otrivial χ or ψ is 1 + G(χ, ψ) q s. For the complex zeros s of this L-fuctio we have q s = G(χ, ψ). Sice q s = q Re(s), sayig the zeros of this L-fuctio satisfy the Riema hypothesis that is, the zeros have Re(s) = 1/2 is equivalet to sayig G(χ, ψ) = q. 3. Euler products ad the Hasse Daveport relatio So far we have used oly the additive represetatio of a L-fuctio, as a Dirichlet series. Usig the multiplicative represetatio, as a Euler product, we will relate the Gauss sum of characters χ ad ψ o F q with the Gauss sum of lifted characters o F q. Theorem 3.1 (Hasse Daveport). For 1 let χ = χ N Fq /F q ad ψ = ψ Tr Fq /F q be liftigs of χ ad ψ to characters o F q. If χ or ψ is otrivial the G(χ, ψ ) = ( G(χ, ψ)). Proof. For ay character η : (F q [T ]/M) C, its L-fuctio has a Euler product: L(u, η) = η(a)u deg A = 1 1 η(π)u deg π, moic A moic π where π rus over moic irreducibles (with η(π) = 0 if π M). Usig the power series idetity 1/(1 au) = exp( k 1 (au)k /k), we ca write L(u, η) as a expoetial: L(u, η) = exp η(π) k u k deg π k moic π k 1 = exp k uk deg π (deg π)η(π) k deg π moic π k 1 = exp dη(π) /d u. 1 d deg π=d We will write the two iermost sums as a sigle sum over the elemets of F q. Each moic irreducible π i F q [T ] of degree d has d distict roots, ad the roots lie i F q whe d. The term dη(π) /d ca be regarded as a cotributio of η(π) /d from each of the d roots of π. For α F q, let π α be its miimal polyomial over F q ad d α = deg π α. The (3.1) dη(π) /d = η(π α ) /dα. d deg π=d α F q Now set M = T 2 ad η(a(1 + bt ) mod T 2 ) = χ(a)ψ(b). This is otrivial sice χ or ψ is. For f(t ) relatively prime to T 2, set f(t ) a(1+bt ) mod T 2. The a = f(0) 0 ad ab = f (0), so b = f (0)/f(0). Thus η(f(t ) mod T 2 ) = χ(f(0))ψ(f (0)/f(0)). If f(t ) = π(t ) is moic irreducible ad π(0) 0 (that is, π(t ) T ), the we ca write χ(π(0))ψ(π (0)/π(0)) i terms of a orm ad trace of a root of π: lettig d = deg π ad α 1,... α d be the roots of π i F q, for ay root α of π we have π(0) = ( 1) d (α 1... α d ) = N Fq(α)/F q ( α) ad π (T ) π(t ) = d i=1 1 T α i = π (0) π(0) = d 1 = Tr α Fq(α)/Fq ( 1/α). i i=1

4 4 KEITH CONRAD Therefore α F q η(π α ) /dα = Replacig α with 1/α, α F q η(π α ) /dα = α F q χ(n Fq(α)/F q ( α)) /dα ψ(tr Fq(α)/F q ( 1/α)) /dα. α F q χ(n Fq(α)/F q (α)) /dα ψ(tr Fq(α)/F q (α)) /dα = α F q χ(n Fq(α)/F q (α)) /dα ψ(tr Fq(α)/F q (α)) /dα = α F q χ = (N Fq(α)/F q (α) /dα ) ψ α F q χ(n Fq/F q (α))ψ(tr Fq/F q (α)), ( ) Tr d Fq(α)/Fq (α) α where the last step uses the trasitivity of the orm ad trace mappigs. This sum over F q is the Gauss sum of the characters χ := χ N Fq /F q ad ψ := ψ Tr Fq /F q o F q, so the right side of (3.1) for our character η mod T 2 is G(χ, ψ ). Therefore At the same time, from Sectio 2 L(u, η) = exp G(χ, ψ ) u. 1 L(u, η) = 1 + G(χ, ψ)u = exp ( 1) 1 G(χ, ψ) u. 1 Comparig coefficiets of like powers of u i these two expoetial formulas for L(u, η) we get G(χ, ψ ) = ( 1) 1 G(χ, ψ), or equivaletly G(χ, ψ ) = ( G(χ, ψ)). This proof of the Hasse Daveport relatio is similar to the proof i [1, Chap. 11, Sec. 4], but that proof uses a multiplicative fuctio λ o moic polyomials that is t a character o ay (F q [T ]/M). 4. Jacobi sums For two multiplicative characters χ 1 ad χ 2 o F q, their Jacobi sum is J(χ 1, χ 2 ) = χ 1 (c)χ 2 (1 c). c F q We will realize a Jacobi sum as the liear coefficiet of a L-fuctio for a character with modulus T (T 1) rather tha T 2. Sice (F q [T ]/T (T 1)) = F q F q by f(t ) mod T (T 1) (f(0), f(1)), a character η mod T (T 1) is a pair of multiplicative characters (χ 1, χ 2 ) o F q : η(f(t ) mod T (T 1)) = χ 1 (f(0))χ 2 (f(1)). Assume χ 1 or χ 2 is otrivial, so η is otrivial. By the reasoig as i Sectio 2, sice T (T 1) has degree 2 the L-fuctio of η as a series i u is 1 + η(t + c) u = 1 + χ 1 (c)χ 2 (1 + c) u

5 L-FUNCTIONS FOR GAUSS AND JACOBI SUMS 5 ad the coefficiet of u here is χ 1 (c)χ 2 (1 + c) = χ 1 ( c)χ 2 (1 c) = χ 1 ( 1)J(χ 1, χ 2 ), c 0,1 which up to the sig χ 1 ( 1) = ±1 is a Jacobi sum. Makig the chage of variables u = 1/q s we ca say η(a) (4.1) N(A) s = 1 + χ 1( 1)J(χ 1, χ 2 ) q s. moic A It s a classical theorem that J(χ 1, χ 2 ) = q if χ 1 ad χ 2 are both otrivial, ad we ca iterpret this as sayig the zeros of (4.1) satisfy the Riema hypothesis whe χ 1 ad χ 2 are otrivial. To get a Hasse Daveport relatio, write the L-fuctio of η as a expoetial i u: 1 + χ 1 ( 1)J(χ 1, χ 2 )u = exp ( 1) 1 χ 1 ( 1) J(χ 1, χ 2 ) u. 1 By reasoig as i Sectio 3, if we set χ 1, = χ 1 N Fq /F q ad χ 2, = χ 2 N Fq /F q the the reader ca check that writig the L-fuctio of η as a Euler product leads to 1 1 η(π)u deg π = exp χ 1, ( α)χ 2, (1 α) u moic π 1 α F q = exp χ 1, ( 1) χ 1, (α)χ 2, (1 α) u 1 α F q = exp χ 1, ( 1)J(χ 1,, χ 2, ) u, 1 so a compariso of coefficiets i the two expoetial formulas for the L-fuctio of η implies χ 1, ( 1)J(χ 1,, χ 2, ) = ( 1) 1 χ 1 ( 1) J(χ 1, χ 2 ). Sice χ 1, ( 1) = χ 1 (N Fq /F q ( 1)) = χ 1 (( 1) ) = χ 1 ( 1), we ca cacel the commo χ 1 ( 1) o both sides ad get J(χ 1,, χ 2, ) = ( J(χ 1, χ 2 )) for all 1. This is a Hasse Daveport relatio for Jacobi sums. Refereces [1] K. Irelad ad M. Rose, A Classical Itroductio to Moder Number Theory, 2d ed., Spriger- Verlag, New York, [2] M. Rose, Number Theory i Fuctio Fields, Spriger-Verlag, New York, 2002.