aa r X i v : . [ m a t h . N T ] N ov Selberg Sums – a new perspective
S.J.Patterson
Abstract
Selberg sums are the analogues over finite fields of certain integralsstudied by Selberg in in 1940s. The original versions of these sumswere introduced by R.J.Evans in 1981 and, following an elegant ideaof G.W.Anderson in 1991 they were evaluated by Anderson, Evansand P.B. van Wamelen. In 2007 the author noted that these sums andcertain generalizations of them appear in the study of the distributionof Gauss sums over a rational function field over a finite field. Thedistribution of Gauss sums is closely related to the distribution of thevalues of the discriminant of polynomials of a fixed degree. Here weshall take this up further. The main goal here is to establish the basicproperties of Selberg sums and to formulate the problems which arisefrom this point of view.
The usual class of character sums known as Selberg sums was introduced in1981 by R.J.Evans, [5]. They were the analogues over finite fields of a class,or rther several related classes, of integrals introduced by A. Selberg in 1944,[12, p.204 ff.]. In fact he had already used an integral of this type in 1941 in[12, p.74 ff.] but was uncertain as to whether the integral was already knownand waited before publishing details.Characteristic of these integrals is that they are taken over a space ofmonic polynomials of a fixed degree and that one factor in the integrandis a power of the discriminant. Selberg regarded the integrals he studiedas a extension of Euler’s beta-function and, indeed, his evaluation gives theintegrals as a quotient of products of gamma functions. The primary exampleis the integral Z · · · Z ( x · · · x i ) α − ((1 − x ) · · · (1 − x n )) β − | Y i 1n the region Re( α ) > 0, Re( β ) > 0, Re( γ ) > − Min( i , Re( α ) i − , Re( β ) i − ). Selberg’sevaluation shows that the integral is equal to i Y j =1 Γ(1 + jγ )Γ( α + ( j − γ )Γ( β + ( j − γ )Γ(1 + γ )Γ( α + β + ( i + j − γ ) . For our purposes a transformation is helpful. This integral can be regardedas over all monic polynomials of degree i with all their roots in [0 , σ , . . . , σ i be the standard symmetric functions in x , . . . , x i . Then one hasd σ ∧ d σ ∧ · · · ∧ d σ i = Y j 1) = − χ = 1 and J (1 , 1) = q − 2. Also J ( χ , χ ) = J ( χ , χ ), J ( χ , χ ) = χ ( − J (( χ χ ) − , χ ) and J ( χ , χ ) = χ ( − J ( χ , ( χ χ ) − ). There are some further relations and properties ofthese sums which have been studied in great detail.Let now k = F q ( x ) and R = F q [ x ]. We define e : k → Q (1 p )by e ( f ) = e o ( P v = ∞ Res v ( f d x ))= e o ( − Res ∞ ( f d x )) . Here, in the sum, v runs through the finite places of k .We define for χ ∈ X q and g ∈ R − { } the Dirichlet character of modulus g f χ ( f /g ) = χ ( R ( g, f )) . 4e define the global Gauss sum associated with k to be g ( r, χ, c ) = X d (mod c ) χ ( d/c ) e ( rd/c )for r ∈ R − { } . This can, by means of the Davenport-Hasse theorem, beevaluated in terms of Gauss sums over F q . If r is coprime to c one obtains g ( r, χ, c ) = µ ( c ) χ ( r/c ) − χ ( c ′ /c )( − τ ( χ )) deg( c ) . Here c ′ denotes the derivative of c . This shows that the function χ ( c ′ /c ) is assubtle a function of c as g ( r, χ, c ). Recall that it is equal to χ ( η (deg( c ))) χ ( D ( c ))for c monic. We therefore have for c coprime to rg ( r, χ, c ) = µ ( c ) χ ( η (deg( c ))) χ ( r/c ) − ( − τ ( χ )) deg( c ) χ ( D ( c )) . In the special case χ = ω , r = 1 we obtain g ( r, ω, c ) = µ ( c ) ω ( η (deg( c ))) τ ( ω ) deg( c ) . As τ ( ω ) is √ q times a fourth root of 1 this gives us the Gauss evaluation ofthe quadratic Gauss sum in this context. One could also use this argumentto prove Pellet’s formula using that evaluation, as given, for example, in [16,XIII, § 11, Proof of Theorem 13].We can now define the Selberg sums which we are going to investigatehere. Apart from a factor ± r to be of the form x e ( x − e . For χ , χ it is X c χ ( r/c ) ω ( D ( c )) χ ( D ( c ))where c is summed over all monic polynomials of a fixed degree i . For ourpurposes it is convenient to modify the expression by using Pellet’s formula.It becomes ( − i X c µ ( c ) χ ( r/c ) χ ( D ( c )) , or, with the reciprocity law,( − i χ ( − i deg( r ) i X c µ ( c ) χ ( c/r ) χ ( D ( c )) . Both of the sums here could be considered as to be entitled to be the morefundamental one. We choose Se ( r, χ , χ , i ) = X c monicdeg( c )= i µ ( c ) χ ( r/c ) χ ( D ( c )) . 5e take χ ( r/c ) to be zero if r and c are not coprime. By means of theDavenport-Hasse theorem we have then Se ( r, χ , χ , i ) = ( − i τ ( χ ) − i χ ( η ( i )) X c monicdeg( c )= i χ ( r/c ) g (1 , χ , c ) . If now there exists an exponent e so that χ χ e = 1then Se ( r, χ , χ , i ) = ( − i τ ( χ ) − i χ ( η ( i )) X c monicdeg( c )= i g ( r e , χ , c )where we restrict the sum to c coprime to r . In this case one can use of thetheory of metaplectic forms to investigate this sums, see [10],[11]. At theother extreme χ = 1 and Se ( r, χ , , i ) is then the coefficient of q − is in theL-series L ( s, χ ( r/ · )) (over R ). It follows from this that if χ ( r/ · ) is non-principal then Se ( r, χ , , i ) = 0 for i ≥ deg( r ) where r is the conductor of χ ( r/ · ) – [1, p.471].We should note that for r of the form θr ao with θ ∈ F × q , a ∈ N one has Se ( r, χ , χ , i ) = χ ( θ ) i Se ( r o , χ a , χ , i ) . In particular we can reduce the calculation of Se ( r, χ , χ , i ) to the case where r is monic. In this section we shall summarize the results of Anderson, Evans and v.Wamelen in a notation suitable for the purposes in hand. We shall take r ( x ) = x e ( x − e . The group X q is cyclic and the notation of these threeauthors is based on the choice of a generator denoted by τ . The parameters a , b and c are related to the parameters here by τ a = χ e , τ b = χ e and τ c = χ . It is convenient to distinguish two cases. If τ a = χ e and τ b = χ e lie in the subgroup generated by τ c = χ then we can define two integers f and f by χ e χ f = 1 and χ e χ f = 1. It turns out that it is this case, theone investigated by v. Wamelen in [15], is precisely the one covered by thetheory of metaplectic forms as formulated in [11]. We shall therefore referto this as the metaplectic case. The alternative, when either χ e or χ e is6ot in the subgroup of X q generated by χ we call non–metaplectic . Thiswas the one investigated by Evans in [6]. This nomentclature is by no meanssatisfactory, nor even accurate, but it has developed from my private usageand I have not found anything better to propose. The distinction does seemto be a useful one as we shall see below.In the discussion below we shall write i instead of n as used by Anderson,Evans and v.Wamelen. This corresponds to the usage in Section 2. We shallalso use n , the order of χ , where they used d . For m ∈ N and j ∈ Z wewrite ( j ) m for the least non–negative residue of j modulo m .In the theory of Selberg sums a particularly important role is played by aproduct denoted by previous authors by P n ( a, b, c ) and which we shall replaceby P i ( e , e , χ , χ ) = Y ≤ j n = 1. For more details see § 5. The mainresult of the three authors is that in the non–metaplectic case Se ( x e ( x − e , χ , χ , i )is equal to χ ( − e i ( − i P i ( e , e , χ , χ )and in the metaplectic case it is equal to χ ( − e i ( − i q [ in ] P i ( e , e , χ , χ )times T (2 (cid:2) in (cid:3) , q ) if ( i ) n ≤ Min( f , f ) ≤ Max( f , f ) < ( f o + f − i + 1) n T (2 (cid:2) in (cid:3) + 1 , q ) if ( f o + f − i + 1) n ≤ Min( f , f ) ≤ Max( f , f ) < ( i ) n S ( (cid:2) in (cid:3) + 1 , q ) otherwise.We note here, following [15, § X m ≥ T (2 m, q ) X m = U e ( q, X )(1 − X ) (1 − q X )where U e ( q, X ) = 2 X q − qX − qX + X + 1 , X m ≥ T (2 m + 1 , q ) X m = U o ( q, X )(1 − X ) (1 − q X )where U o ( q, X ) = X q + q X − qX − q + 2 , and X m ≥ S ( m, q ) X m = 1 + ( q − X (1 − X ) . We note also that U e ( q, q − ) = (1 − q − ) , U e ( q, 1) = 2( q − , U o ( q, q − ) = − q (1 − q − ) and U o ( q, 1) = 2( q − . These results mean that in the non-metaplectic case and in any complex embedding we have X i ≥ i ≡ i S ( x e ( x − e , χ , χ , i ) X ( i − i ) /n 8s equal to χ ( − e i ( − i P i ( e , e , χ , χ ) / (1 − χ ( − e n ( − n AX )where we assume that 0 ≤ i < n and A is as above. Note that | A | takes on,if n = 1, one of the three values q n − , q n − or q n in any complex embedding.If n = 1 then | A | takes on one of the three values 1, q or q in any complexembedding.In the metaplectic case we find that, according to the three cases above,the series is equal to χ ( − e i ( − i P i ( e , e , χ , χ ) U e ( q, χ ( − e n ( − n AX )(1 − χ ( − e n ( − n qAX ) (1 − χ ( − e n ( − n q AX ) ,χ ( − e i ( − i P i ( e , e , χ , χ ) U ( q, χ ( − e n ( − n ) AX (1 − χ ( − e n ( − n qAX ) (1 − χ ( − e n ( − n q AX ) , or χ ( − e i ( − i P i ( e , e , χ , χ )(1 + ( q − χ ( − e n ( − n qAX ) (1 − χ ( − e n ( − n qAX ) respectively. Note that the nature of the singularities reflect the type of theSelberg sum. As we have assumed that n > | A | = q n − . We shall now turn to a property of Selberg sums analogous to Theorem 1 of[11]. We need some preparations in order to be able to formulate the result.Let n be the order of χ and let n ′ be the order of χ . Let χ be such that χ and χ are in the group generated by χ . Let a ≥ b ≥ χ = χ a , χ = χ b . Let π monic and a prime in R = F q [ x ]. If π | r then Se ( π n ′ r, χ , χ , i ) = Se ( r, χ , χ , i ) . If π r this is no longer true. We have that Se ( r, χ , χ , i ) − Se ( π n ′ r, χ , χ , i )is equal to X deg( c π )= i µ ( c π ) χ ( r/c π ) χ ( D ( c π )) . D ( c π ) = D ( c ) D ( π ) R ( c , π ) this becomes − χ ( r/π ) χ ( D ( π )) X deg( c )= i − deg( π ) µ ( c ) χ ( r/c π ) χ ( π/c ) χ ( D ( c ))or − χ ( r/π ) χ ( D ( π ) Se ( r a π b , χ , χ , i − deg( π )) . We regard this as a stability property of Selberg sums. Theorem 1 Let α, β, γ, δ ∈ F q be such that ∆ = αδ − − βγ = 0 . Then forsuitable integers M, M ′ we have that χ (∆) − i Se r (cid:18) αx + βγx + δ (cid:19) a (cid:18) γx + δ ) (cid:19) b ( i − M , χ , χ , i ! is equal to Se ( r ( x )( − γx + α ) M ′ , χ , χ , i ) If γ = 0 we can take M, M ′ = 0 . If γ = 0 then M is to such that χ M = 1 and M + b ( i − > a · deg( r ) , and M ′ is such that M ′ > and χ M ′ = 1 .Proof: The proof is carried out by verifying the identity for (cid:18) α βγ δ (cid:19) ofthe form (cid:18) λ (cid:19) , of the form (cid:18) α δ (cid:19) and equal to (cid:18) (cid:19) and thencombining these in the usual manner.The first two cases are straightforward; we need only replace c in the sumdefining the Selberg sum by c ( x + λ ) and c ( θx ) θ − i where θ = α/δ respectively.In the case (cid:18) α βγ δ (cid:19) = (cid:18) (cid:19) we assume that r (0) = 0; this is thereason for the introduction of the parameter M ′ . This being so c is notdivisible by x and so c (0) = 0. We replace c in the sum by c ( x − ) x i /c (0) andrecall that R ( x, c ) = c (0). The formula quoted now follows from the theoryof resultants. The results of Section 3 show that there is a large number of relationshipsbetween Selberg sums. The results of Anderson, Evans and v. Wamelen showthat polynomials with two rational zeros over F q can be evaluated explicitly.It turns out, as we shall see in the next section, that if the zeros are no longerrational a similar formula holds. The next case which one can investigate, asin [11] is that of polynomials with three rational zeros. In view of Theorem10 these can be brought to the form x e ( x − e ( x − λ ) e λ . As in [11] thereare a large number of relations between these, now a little more complicatedas we have to move into the region of stability. They are the analogues ofthe relations beween such sums similar to transformations of hypergoemetricfunctions (cf. [17, Chap. XIV]). We should note that in this standard formsuch Selberg sums can be easily computed as the resultant can be evaluatedby a continued-fraction type of recursion. This is implement in, for example,the gp/PARI package. I hope to discuss the results of these calculations ina future paper.There are some further relations that are of interest. Let n, n ′ be as inSection 3. If n ′ | n then the generating series X i ≡ i (mod n ) i ≥ Se ( r, χ , χ , i ) T ( i − i ) /n which converges in | T | < q − n , is, for each i : 0 ≤ i < n , a rational functionin T . There is at most one singularity in | T | < q − n/ and this, if it exists, issimple and is located at T = q − ( − τ ( χ − )) − n . These are the series investi-gated in [10] and [11]. The determination of the residue and the establishingof its properties is one of the major questions addressed in those papers andit is one that has, as yet, only been partially answered.If the condition n ′ | n is not satisifed then the series above can also beinvestigated by means of the theory of Eisenstein series. The point here isthat one has to use an Eisenstein series of a “Nebentypus” depending on χ and χ . The general theory in contained in [8]. In fact in this case theconstant terms of the Eisenstein series will be made up from holomorphicL-series. The analytic continuation then follows from the “principle of theconstant term”; see [9, Theorem 4.8] for a statement and [7, Theorem 1.6.6]for a proof in the function–field case. The details have not been given explic-itly but it seems very plausible that in this case one will be able to concludethat the series above also in this case represents a rational function but onenow with no singularities in | T | < q − n/ . One expects that one will be ableto determine the denominator and to estimate the degree of the numeratorwhich are known in the metaplectic case also in the non-metaplectic case.In is instructive to examine the special case n = 1. There are twocases, the metaplectic case in which χ ( r/ · ) is a principal character andthe other case when χ ( r/ · ) is non–principal. In the first case we see that P i ≥ Se ( r, χ , χ , i ) T i is L ( s, χ ( r/ · ) − with T = q − s . Let r be the modulusof χ ( r/ · ); then this series is equal to(1 − q − s ) Y π | r (1 − q − deg( π ) s ) . 11n this case the generating series is a polynomial.If χ ( r/ · ) is non–principal then, if we suppose for the moment that the χ ( r/ · ) is primitive. Then Anderson, [1], has observed that L ( s, χ ( r/ · ) is apolynomial in q − s of degree deg( r ) − 1. It follows that L ( s, χ ( r/ · ) − has thisnumber of singularities, counted with multiplicities. They lie on q − s = q − / by Weil’s theorem. If the character is not primitive then there is a numeratorof the form Q (1 − q − s deg( π ) ) where the product is over all the primes dividingthe modulus but not the conductor. In the case of the standard Selberg sumsthe degree of is 1 and it is of the form 1 − αq − s where α is a Jacobi sum.The case of other r will be much less straightforward.We return to the general case. The singularities of the generating seriesof the sequence Se ( r, χ , χ , i ) are given as certain sums of their coefficients,i.e. of Selberg sums. This has been described in [10],[11]. The theory ofmetaplectic groups then leads to a number of relations between the residues,the so–called “Hecke relations”. A consequence of this is that there are anumber of unexpected relations between Selberg sums. Some examples aregiven in [10, § R . We shall now sketch his technique. Let f be a monic polyno-mial in R . Let χ be as before and let n be its order. We regard χ ( f / · ) as aDirichlet character. Let f o be the conductor of f . This is Q π where π runsthrough those monic prime divisors of f whose order in f is not divisible by n . We shall assume that f o = 1. Then by [1, Prop. 2.1] ω ( D ( f o )) χ ( f /f ′ o ) τ ( χ − ) deg( f ) − Y π | f o τ ( χ − ord π ( f ) ) − deg( π ) is equal to X g :deg( g )=deg( f o ) − g monic χ ( f /g ) . This is proved by using the functional equation for the L-function L ( s, χ ( f / · ))and the fact that the latter is a polynomial to determine the coefficient of( q − s ) deg( f o ) − . Anderson proves this in the case n = q − χ ( f / · ) is not principal. Anderson achieves the same generalitythrough his parameter c .Anderson’s proof of the formula for the Selberg sum exploits the eval-uation of a double sum in two different ways. The argument is strongly12eminiscient of the the multiple Dirichlet series technique, as used, for exam-ple, in [3]. The crucial point of the argument is that r o should be quadraticand it follows that one can also evaluate Selberg sums explicitly when r is apower of an irreducible quadratic polynomial.It seems plausible that one could use Anderson’s method combined withelementary considerations to determine the properties of metaplectic Eisen-stein series over rtional function fields. In fact the “principle of the constantterm” referred to above is relatively elementary and as the constant term iseasy to study in this case even the Eisenstein series approach is relativelyelementary. The method of multiple Dirichlet series is, at least for higherranks, an important component in the study of Eisenstein series and so theapplicibility of both methods should not be surprising. At any rate it shouldreassure those unfamiliar with this theory that the method is, at heart, inthis case at least, elementary. References [1] G.W.Anderson: The evaluation of Selberg sums, Comptes RendusAcad.Sci.Paris,Ser. I , 311(1990)469-472.[2] D. Carmon, Z.Rudnick, The autocorrelation function of the M¨obiusfunction and Chowla’s conjecture for the ration function field, Quart.J. Math. Acta Arith. Numberfields and function fields – two parallel worlds Birkh¨auser,2005, 71–86.[5] R.J. Evans: Identities for products of Gauss sums over finite fields, L’Enseign.Math. L’Enseign.Math. Ann.Math. 100 (1974) 249-306.[8] D.A. Kazhdan, S.J. Patterson: Metaplectic Forms, Publ.Math. IHES Can. J. Math. Glasgow Math. J. in preparation [12] A. Selberg, Collected Papers, 1 , Springer, 1989.[13] R. G. Swan, Factorization of polynomials over finite fields, Pacific J.Math. Modern Algebra, Vol. 1 , (Trans. F. Blum)Fredrick Unger Publishing Co.,New York, 1949.[15] P.B.van Wamelen: Proof of the Evans-Root conjectures for Selberg char-acter sums, J.Lond.Math.Soc.,II.Ser. Basic Number Theory , Springer, 1967.[17] E.T.Whittaker, G.N.Watson, Modern Analysis , 4th edition, CambridgeUniv. Press, 1946Mathematisches InstitutBunsenstr. 3–537073 G¨ottingenGermanye-mail: [email protected]@gwdg.de